Head towards the exit door which you used previously. Unfortunately the passage will be blocked by the Penguin #1 and as a result you will have to use the Gotham codes. Stay beside the exit and turn on the Detective Mode. Look for a grate in the floor #2 and press the right trigger and A to enter it.
Use the shaft to reach the locked-off room in the north #1. Get out of the shaft (A) and take a look around the area. Choose the Cryptographic Sequencer, hold down the left trigger, aim at the door panel #2 and press the right trigger.
You will have to solve a new type of mini-game, though it's not all that complicated. You need to use the analog stick to create a password. I'd recommend starting off by moving the left stick (clockwise, or counter) until the device stops on the first part of the password #1. Afterwards do the same with the right stick which will create the password DISSECT #2.
Once again jump into the shaft and return to the main corridor. Now you will be able to use the door leading outside #1. Here use the Electrical Charge once again to set the industrial engine into motion #2. Afterwards run onto the gate and slide underneath it.
Now you can head to the local Museum. The building can be found in the south-west part of Arkham City and you need to get to area of the main entrance #1. Note the Thugs gathered by the gate #2. It would be good to initiate the fight with a surprise attack and then finish them off conventionally. You shouldn't have a hard time fighting them, though you need to look out for the ones carrying weapons.
Afterwards approach the entrance and use the Cryptographic Sequencer on the right control panel #1. Solve the mini-game by moving both analog sticks and create the password FOSSIL #2. Now you can use the door and enter the Trophy Room.
After getting inside you will be forced to fight a new type of enemy, Thugs equipped with knives #1. If you want to test out the suggested tactic, push the analog stick back and press Y at the right moment. It will make Batman automatically dodge incoming attacks #2.
The tactic described above is unfortunately rather risky and not particularly effective when there's more than one enemy with a knife. As a result I'd suggest jumping over the enemies by leaning the analog stick towards the enemy and pressing A #1. Regardless of the chosen tactic, you have to defeat them both #2.
If you turn on the Detective Mode, you should note that there's a laser barrier #1 which blocks access to the main part of the museum if crossed. In order to solve this problem, go right from the entrance and jump through the small window #2.
Approach the control panel found in this room #1 and prepare the Cryptographic Sequencer. Unfortunately you won't be able to hack it, as the Penguin has placed jamming devices in the area #2. As a result you will have to find and destroy them.