Make sure that you're standing in a correct spot, so you'll have a wall behind you. Wait for one of the monsters to begin moving your way and throw a Batarang at it. Follow it by jumping to the side and the mutant should hit the wall. This is your chance to throw a few punches, however watch out for the other mutant at the same time and prevent it from injuring Batman.
I would recommend that you focus on inflicting injuries to both mutants and not only one. Thankfully their health bars can be seen in the upper right corner of the screen. You can also use an additional trick when you notice that one of the beasts has kneeled down. Approach the monster quickly and press the action key to climb on its back. You can now control the monster (not fully) and you should use this opportunity to harm the other monster.
Once you've eliminated the first mutant you should go back to your standard pattern, using Batarangs and throwing quick punches to inflict some minor injuries. It shouldn't take too long for you to neutralize the second beast as long as you're careful. Proceed towards a nearby control panel and use the sequencer to destroy it.
You may now approach a large cap. Listen to Batman's statement concerning his inability of reaching the other side of the abyss. Wait for Batman's jet to drop a package into the factory and press the action key to take it. Doing this will allow you to acquire a new gadget. It's going to be a Line Launcher.
Turn around and use your newly acquired toy to get to the other side safely. Remember to aim directly at the door, not at the surrounding walls. Return to Aviery and then head back to the Flooded Corridor.
You'll now have to return to the area where you've rescued Jordan the last time you were here. Proceed to your right and approach another large gap. You'll have to use the Line Launcher for the second time to get to the other side. Proceed to your right and open a nearby door. A short cut-scene will feature Poison Ivy, however you won't fight her at this moment of the game.
Once you've returned to the Flooded Corridor go back to the large gap. There are some inmates in front of you and you can attack them using Batarangs or take on directly after using a Line Launcher. It shouldn't take too long for you to eliminate them. After you've won return to the ventilation shaft and go back to the Abandoned Chamber.
Return to the Flooded Corridor the same way you came before. Notice that the central area is now electrified again, so you can't reach the other side on foot. There are some inmates in front of you. Attack them from here (using Batarangs) or after you use a Line Launcher to reach their positions. Once they've been pacified head on to the Botanical Glasshouse.
Five new inmates have appeared in the area and they're all heavily equipped, so it won't be easy to succeed. Start off by performing a surprise attack on the nearest guard (screenshot #1). Once he's gone start using gargoyles and eliminate the thugs one by one. Don't forget to use Glide Kicks, Inverted Takedowns and other stealth actions.
Once you've won return to the Glasshouse Entrance area. Start moving forward, however you'll soon notice that Poison Ivy's roots have blocked the main path. Thankfully it's not a major problem, because there's a small hole in the floor. Choose the tunnel and you'll soon be allowed to use your hook. Open the main door to return to Arkham East.