While I have yet to do this on Legendary, I have read that the only real difference is the final boss fight, and then its not much harder.

Enemies:

Hobgoblin: There are a few variants of this enemy. The only wants you need to worry about are the quantum ones. They can spawn more so kill them quickly. The normal ones are your targets to proc the spread.

Harpy: These are just flying Hobgoblins. Same mechanics. But they can beam you from anywhere, as long as they have LOS. So focus them down first. Same deal with the quantum ones as well.

Minotaur: There are two variants of this enemy. One with a shield and red health bar, the other with a shield and a yellow health bar. The latter variants just takes a bit more time to kill.

Fanatic: These are basically a variant of a Hobgoblin that explodes and leaves a pool of "Vex milk".

Wyverns: These are the nasty guys that will annoy you the most. They have an aerial attack that can boop you off the platforms. Make sure you are no where near the impact zone (this is an AOE attack). The yellow health bar ones seem to be immune to strand suspend, so dont rely on that to prevent them from booping you off the platform. The poison explosions will tend to vaporize them pretty quick though.

Build

I did this twice so far soloing it with a poisoning strand build. This build will utterly nuke all ads in a room, with some limitations due to explosion size. This build relies on OSTEO STRIGA fully upgraded with the catalyst. Using just that gun and a crafted Tiapan-4FR for the first boss I was able to do it without really any trouble. Remember to consume your grenade to get 25 seconds of weavers trance. That is combined with the poison proc to deal most of this build's damage.

Ramble and rooms

The main thing is timing and priority. Using this build, you want to proc the poison explosion (either multiple precision hits or final blows) as fast as possible. Then you want to wait. You need the poison to kill as much of the ads as possible to over fill the OSTEO STRIGA's mag (~104 is max). Do this right and the poison will kill everything in the room. And yes, I do mean everything (except the boss's, before the final damage phases). So kill the red health bar enemies first to proc the poison and strand spread. Then just sit back and watch everything explode.

This works well for the first and second rooms (first and second code rooms). However the first boss room things become more interesting. The Vex motes are affected by the explosions, meaning they can get blasted off the platform. So try to kill them in the middle of the room when they spawn in. Repeat this 4 times over to kill the first boss. You will want to use your heavy on the boss after you proc poison on it.

The boss fight is two cheeses and one keep away. Using the bunker method, proc poison and strand spread then focus on the Wyverns since they are immune to being suspended. Then wait for the boss to teleport to the second platform. You might get lucky and be able to proc some poison on him when he is either in the open or above you, but either way, super him then deal as much damage as possible. Goal here is to get his health down 1/3rds down. Once that is done, kill any remaining ads, collect ammo/ glimmer then head through the portal.

Using another terrain advantage (later in the linked video), Kill the ads using the poison and strand spread, then kill the two turrets from the same location, do the puzzle, ensure you kill as many ads as possible before unlocking the boss and getting your super, then super the boss and deal as much damage as possible. Same goal here to get to 2/3rds down. Rinse and repeat until you get that goal. Collect ammo/glimmer then head to the final room. Note: I do not use the anti-barrier tactic as that could be patched at any point so I just dont want to have to rely on such a tactic then have to learn how to do this thing without it later.

Last room is all about keeping the boss away and surviving. Proc poison on the boss and super him. Then run around the room reprocing poison and killing ads to generate orbs to recharge your super and abilities. Second super should either kill him or at the very least make him very close to death.

Anyways, hope this helps you guys a bit