The First Room
In the first room, the one with the hologram of the Heretic Leader, there are five combat flood forms on the ground. You should firstly sword all of these bodies, so that they don't get revived. Killing the two allied Grunts saves time by skipping dialogue, but you should take care to avoid aggravating the Elites. One way to do this is to kill one Grunt immediately as the level begins, and then to kill the other immediately after the Heretic Leader hologram appears. Next, pick up a new sword and pick up a sentinel beam (both found on the dead bodies of the flood combat forms). Dropping your carbine near the exit door is recommended; this way when the door opens you can swap it with your sentinel beam without losing any time. Afterwards, jump up onto the pillars, or further up into the rafters of the room. It is about to flood (literally) with infection forms. The waves can spawn in any of the various holes around the upper walls. Two initial waves spawn in these holes, then two in the center, then two in the holes again, then two in the center followed by a final wave from the holes. Be ready by the door as the final wave spawns.
The elevator is an auto-scroller as far as anyone knows. So long as you avoid death, there is no real timesave or timeloss for the first few minutes. The flood combat forms can be killed from afar before they reach the elevator, and a lot of the time the real threat is posed by your elite allies, who can melee you to death and stick you. Punching them off the edge or just flat-out killing them is advisable for this reason, though it should be noted that the second one may be aggravated. Eventually, there will be a wave of sentinels that spawns. A canister of gas on the elevator floor will begin to rise, and if you watch it you will see it change color when it emerges fully. At this precise moment, the kill-floor beneath the elevator despawns and you can drop down to the platform to continue the level (be sure to drop down on the correct side, with the canister on your left if facing the center of the elevator). The audio cue that plays shortly afterwards can alternatively be used as timing. You have to drop down in two stages to avoid dying to fall-timer. It will likely take some practice to consistently drop down without bouncing off into empty space.
Sword cancel down the hallway after the elevator and you will be in the Flood/Hell room.
This room takes some practice to execute correctly. There is a Sentinel Beam by your feet as you enter, which you should pick up before dropping down. The initial wave of enemies that are already in the room include a number of Carrier Forms, two Heretic Grunts on turrets, and one Heretic Elite. Also, there are dead Combat Forms on the ground. It's recommended to take care of the Grunts and Heretic Elite first, destroying the Combat Form bodies as well if you can. Next, kill the Carrier Forms with the sentinel beam as fast as possible. Take note that when you kill the carrier flood there will be a lot of infection forms running around. After you eliminate all those enemies mentioned, the next wave will spawn. This wave will include three Heretic Elites, who enter via the same door that you do, and Carrier/Combat Forms that spawn near the roof of the room. It's recommended to kill the Elites first, by catching them as they enter the room; you can jump back up to the upper level through any of the windows if you first jump onto one of the ledges that sit between them. The flood will sometimes sit up in the rafters for some time, so be sure to check that they've all jumped down or been killed. After you kill all of them, the music will change, and for the next minute exactly the room will continually spawn more combat forms as you kill them, keeping the total at four. It's therefore futile to kill more flood during this time. Instead, it is best to jump up onto the small ledge atop the exit door, herding the combat forms to that area. Keep an eye on your timer, and kill them as soon as the sixty seconds have elapsed. The door will then open, with Heretic covenant on the other side. Kill these as quickly as possible and wait in the airlock for the door to lock behind you. It is not necessary for all enemies to be dead; if they are sufficiently far from the airlock it will lock anyway.
The Second Elevator Room
Make your way across the open bridge, heading either left or right (there's a nice swordfly towards the right side, available off of the closest Carrier Form) to head towards the next room. There are three Sentinels and four Combat Flood Forms in this room (usually). Kill them as fast as possible and the screen will fade to white. Skip the cutscene and you are now tasked to go up the elevator and cut the cables at the top. There are shortcuts to get to the elevator faster, using a combination of crouch jumps and optionally a sword fly. Once you get to the elevator and go to the top, there will be three cables to cut using your Energy Sword (or alternatively by shooting them on their outer face). The order you cut the cables in doesn't matter. You can cut the first cable as soon as you can get to it, but the next two can only be destroyed once Halfjaw has finished his respective line of dialogue. Once all three are cut, the Combat Forms and Sentinels will turn passive and pose no further threat. Head back to the elevator in the center and hold X while facing the panel to begin the ride back down as soon as possible.
As you travel back down, look slightly to the right, at about a one-o-clock orientation from the panel at the top. Drop down to the bottom level and continue through the door in front of you. There's no need to call up the elevator here; you can just drop down, but be careful as the fall-timer will kill you if you don't walk straight off. In the next room, you can either attempt a sword cancel across (spawns permitting), or simply go around to either side. A cutscene triggers once you pass through the door on the opposite side, but be careful as you can often die to a sword in the back from two Heretic Elites that spawn in there. A very brief Banshee section ensues. The flying debris from below can kill you, so it's wise to fly over the beam near the far platform. Continue through the level normally, backtracking through the rooms of The Arbiter in the opposite direction. Eventually you'll find your way back to the hangar room and will fight the Heretic Leader.
Skip the cutscene that triggers as you enter, and move to the right side of the Seraph Fighter. The real Heretic Leader is the one furthest from you when you start. Kill him as quickly as you can, using sword lunges, a Plasma Grenade, or your Carbine. The two at the front are holograms and will vanish when the real Leader is dead.
Playing through this mission on Legendary is largely similar, with some notable differences. First and foremost, there is obviously a much greater risk of dying and losing time at essentially every part of the mission past the initial room.
The Elevator has to be played more cautiously. Keeping your allied Elites alive is more advisable to reduce the likelihood of being overwhelmed by Flood, though the same risks of them killing you apply. It can help to swap the one carrying a single Plasma Rifle for a Carbine, as he'll be much more effective with it. The other one will throw grenades during the elevator ride if you give him a Carbine, so this is best avoided (he otherwise does not, as he is dual-wielding). If you start running low on Carbine ammo, keep an eye out for Combat Forms carrying a Carbine and try to take the ammo from them.
The order in which you kill the enemies in The Flood Room is the same. It is very important that you jump up to the Heretic Elites that enter via the entrance door as soon as possible, as they pose little threat to you in close quarters (their melee attacks are unlikely to damage you in most cases). If they spread out or head towards the turrets in the middle, you're losing time and are in much greater danger of dying. Instead of waiting above the airlock door for the sixty seconds, jump up through one of the windows and wait up there. The Heretic reinforcements in the airlock can be killed quite quickly and consistently using a well-placed Plasma Grenade or two, followed by sword lunges and Carbine headshots.
The Second Elevator Room
The aforementioned swordly off of the closest Carrier Form is important on Legendary; you're unlikely to be able to simply run past everything without dying. Strategic camouflage usage is needed to safely get by the Heretic Elites that spawn on your way to the next room, and in the next room as you make your way up to the elevator. Cutting the cables plays out mostly the same, just take care not to be melted by Sentinels or melee'd by Combat Forms.
If you avoid killing anything in the first room of The Escape, there is typically not much to worry about in the next (although there is often a Heretic Elite and/or Combat Form in there). A tight crouch jump up to the ledge on your left as you enter is advised, as it is slightly faster and somewhat safer. The hallways following this can get a bit scary, with various enemies spawning in from doors to the side. Ideally, you should swordfly past the Carrier Forms to the Heretic Elite running from you; if you kill him and the Grunt in the next hallway in time you stand a good chance of getting a Checkpoint before the next room, which can go very, very wrong. The typical safe way to approach this room is to head up the ramp to the left and jump across the railings to the far end, much like in The Arbiter only reversed. To be really safe, you can instead walk across and to the right of the room, and wait there for more camo to continue. Be wary of the explosive canisters lying in the center of the room, as they can easily kill you. It is best to shoot them first. Lastly, you can attempt a swordly directly ahead on the Heretic Elite that (usually) is up on the far platform. Hit A mid-flight. If you do this well you can sometimes avoid spawning the two needler Grunts in the hallway heading out of the room.
The area that is most different on Legendary is the Heretic Leader fight. Instead of simply dying the first time you attack him, the Heretic Leader will fly up to one of the four tubes on the walls (the same ones that Sentinels appeared from in The Arbiter). From these tubes he delivers three different pieces of dialogue, and during each one he is passive and invincible. After each bit of dialogue he spawns more holograms and tries to resume the fight with you in the middle of the room. You should disallow this by swordflying to him each time, and swording him into submission as he finishes the dialogue. It is best to attack his side or back, as unlike other Heretic Elites his melee is typically an insta-kill and he can perform one if you attack him from the front. After each repetition of this he will fly up his tube and appear again, sometimes in the same one. He can get stuck whilst doing this, if this happens just sword-lunge him and he should go up. If he takes too long to do this on the second one in particular, you will sometimes get additional dialogue from the Oracle that loses time. Another complication can arise if you knock him out of the tube; he will not fly back up once you deal enough damage to him if he is too close to the wall. If you do this, just push him closer to the center of the Hangar using your sword and he should resume as normal. On the third and final bit of dialogue, drop his shields with the sword, and back off to shoot him in the face with your Carbine as his dialogue ends. He will now finally die and the level will end.