Ignore the fact that you'll be observing your surroundings from the first person perspective, because everything is proceeding according to the plan. Approach the glass wall and listen to Joker's statement. Once the explosion has occurred enter the next room and proceed to your right. Watch a new cut-scene and get ready for a battle to take place in the Throne Room.
This is going to be an extremely challenging fight during your which you'll not only have to defeat two mutants but also a lot of Joker's thugs. Standard inmates will keep appearing constantly, so you won't have too much time to rest. Another important factor is that the arena you're fighting in is going to be small, so it'll make dodging enemy attacks more complicated.
During the initial phase of this battle you should focus your attention on avoiding mutant attacks by performing successful jumps. Make sure to stay close to the inmates, so there's a chance they'll be hit by the beasts. You can also encourage the monsters to smash into one another. When you have a free moment attack the inmates, but each time make sure you're not in any danger from the mutants.
Once you've been able to defeat most of the inmates you can start using Batarangs to stun the beasts. Make sure to do this only when they're running towards Batman. After each hit you may want to attack the stunned beast, however you'll have to watch out for the other mutant at the same time. Don't forget that your primary concern should be to keep Batman alive.
Sooner or later you'll be given a chance to climb on the mutant's back. Use this opportunity to defeat as many inmates as possible and to inflict major injuries to the other monster. Don't forget that defeating both mutants is the key to your success. Once you've returned to the ground continue performing all the actions mentioned above.
Once you've been successful in defeating the first mutant it should become much easier to stay alive. Don't forget to use Batarangs and watch out for the inmates. Neutralizing the second mutant as well will allow you to advance to the final battle with Joker.
The action of this second battle will take place on the roof of the Penitentiary building. Once the battle has begun start running around Joker and avoid his claws. Thankfully your opponent won't even try that hard to get you. Keep running until Joker decides to climb on top of a nearby balcony. You'll advance to the first (and easiest) phase of the battle.
Joker will now send his goons to finish you off. I assume you've mastered your fighting skills by now, because otherwise you'll have a lot of problems winning this entire battle. Keep attacking Joker's men. Don't forget to perform successful countermoves and cape attacks. You should also focus on using your special combo moves in order to dispose of enemy units quickly.
The only real problem during this first phase of the battle are going to be Joker's teeth. They're rigged with explosives, so you'll have to be very careful while moving on the arena. Whenever you hear a cruel laugh perform an evasive maneuver, because an explosion will occur in about a second. It would be a good idea to wait for the teeth to show up near enemy units, so the explosions will stun them.
Once you've eliminated all the inmates Joker should turn around, because he'll be distracted by a helicopter. MAKE SURE that there aren't any additional thugs in the area. The same goes with Joker's teeth. It's extremely important, because otherwise you wouldn't be able to end this phase properly. Quickly choose an Ultra Batclaw from your inventory and fire it at Joker's back. You'll now have to keep pressing the action key rapidly to pull him from the upper balcony. If you're interrupted you would have to remain in the same phase and eliminate a new (smaller) group, so that's why it's important to secure the area before focusing on Joker.
If you've done everything according to the plan you'll watch a short cut-scene showing Joker's fall. Approach the boss quickly and perform a standard 3-hit combo. Joker should lose 1/3 of his initial health. Once again you'll have to avoid his direct attacks which shouldn't be a problem. Once Joker has returned to the upper balcony you'll advance to phase two.
The only real change during this second phase of the battle is that aside from standard inmates you'll also be dealing with thugs armed with stun guns and knives. Obviously you should focus your attacks on them. Use cape attacks to disarm enemies with knives and jumps to outsmart goons with stun guns.
Don't forget about the bombs, especially since you'll be dealing with even more teeth than the last time. Once you've neutralized ALL enemy units wait for Joker to turn around, use the Ultra Batclaw on him and throw a few punches. Before you'll advance to the last third phase you'll have to avoid his claw attacks again.
The third phase of the battle will feature one new element - a gun locker. It's crucial that you prevent your enemies from opening the locker. It'll be located directly beneath Joker's balcony. If they do somehow manage to do that make sure that the gun remains on the floor, because otherwise you'd be facing a serious threat.
There are also electrified barriers near the locker, so you'll have to be careful not to touch them. On the other hand you may try pushing the inmates towards the barriers to make your job easier. Once you've eliminated all the inmates and allowed all the bombs to explode grab Joker for the last time and pull him from the balcony. This time you won't have to finish him off yourself.
That's it! Sit back and enjoy the ending. Congratulations!