Head back to your apartment. Once you get there, drop into the cage like you did before and approach your window. Then a fight will start. They are a pretty simple group of enemies. Take them out and head to the museum. When you get to the museum, there will be a few guys, one with a shield, in front. Deal with them, head inside and deal with the group in the dinosaur exhibit. A couple of them will have knives.
Run down the hall, through the door and into the gladiator room. There's a good number of enemies here, a couple with shields and knives and lots of hammers and thrown weapons. Thankfully there are no Titans or guns so just keep moving and using your takedowns and you should be fine.
Go through either door, up the stairs and through the door at the top. You'll be in the torture chamber now. You want to swing under the catwalk and head right. Climb through the grate into the caged room. Then you want to pounce up to the catwalk above and follow that path around to the north wing towards the armory. You can drop down and there will be a few enemies just down the hall. Be fast because they are packing guns but thankfully there's only three of them.
Head into the armory to start a stalker section. This section is interesting because you can't get rid of all the guards. They keep respawning. Your only real target is Two-Face. Depending on where he is, you need to sneak up on him in a different way.
If he's on the bridge (like when he starts), you need to get over and hang on the ledge and wait for a ledge takedown. If he's by the entrance walking between the two balconies then you need to wait for a clear shot to sneak up behind him. You can also catch him on the lower level. He will walk around the Mammoth in the center. It's rather hard to catch him here but if you can get around behind him for a silent takedown, it'll make finishing the fight a lot easier.
The best method I've managed is to go to the back of the room so you're near the Mammoth's rear end. Climb up the ledge of the bridge so you are on the right side and wait to do a ledge takedown. Once you've done it, drop down, turn 180 degrees and then run forward and left into the alcove. There should be a vent here. Climb through it, take a left and run to the end and climb up into the vent. Come out the other side and try to make your way into the room and back up to the Gargoyles. With any luck, Two-Face will go back up to the bridge and you can rinse and repeat. Feel free to use a pouncing attack if that is what is convenient (for instance, when he's on the bottom floor).
One you've finished with him, you are back to Batman and can roam the city completing riddles and side-quests.
