Start off by heading forward. You'll want to act quietly here, because there's an armed guard nearby. Make sure to hold the crouch key, so he won't hear you coming. Once he's been eliminated take care of his colleagues. Winning shouldn't be a problem and once the fight has ended proceed towards the lockers. This passageway will be located to your left (screenshot #2).
You won't encounter any additional enemy units in your vicinity, so you won't have to make any stops along the way. Eventually you should be able to reach a passageway leading to the next area (screenshot #1). Wait for Batman to be scanned and enter the Main Cell Block. You should use your detective mode here to continue following the trail. You'll be lead to the southern passageway.
Continue moving south and it shouldn't take too long before you enter The Green Mile. Head on to the main room where you'll be rewarded with a cut-scene featuring Poison Ivy. Ignore Ivy and choose the southern corridor. This time focus on finding a door leading to the Security Control Room.
Once you're inside a new room you may as well ignore a man seen behind the glass wall, because he's a shapeshifter. Instead use the stairs to get to the top of the security tower. Approach Quincy and press the action key to rescue him. In return you'll unlock a very useful gadget - a Cryptographic Sequencer.
It shouldn't come as a big surprise that you'll have to test this new gadget right away. Approach a nearby control panel and use the sequencer. The idea of this mini-game is to make the display of the sequencer turn bright green and to keep it that way for about a second. This will hack the mechanism, making the panel explode and unlocking the passageway.
Leave the guard tower and return to The Green Mile. You'll have to be very careful, because you're going to be attacked by an insane convict here. If you don't want to fight him (you would have to start pressing the action key rapidly) press the counterattack key when a blue icon appears over his head. Don't forget to finish him off with a takedown!
Head on to the Main Cell Block area. Make a few steps forward after your arrival and enjoy a new cut-scene involving Harley Quinn and other inmates. Once the cut-scene has ended QUICKLY use your hook to reach one of the upper platforms. Start attacking the inmates, however bear in mind that some of them will try accessing gun lockers. They have to be eliminated before they get their hands on guns.
Once you've taken care of the immediate threats you may focus on attacking other inmates, however prevent them from picking up weapons from the ground. Watch a new cut-scene when this is all over. Head on to a nearby control room. You'll have to turn left and use the newly acquired sequencer on one of the control panels.
You'll now have to return to the ground floor. This can be done either by jumping over a railing or by hacking into yet another control panel located near the stairs. Once you're there find a small passageway (screenshot #2) leading to the western section of the cell block. Watch out for other insane inmates and use tactics mentioned earlier on to take them out.
Keep heading west and eventually you should be able to find the stairs. Use the stairs to travel to the Guard Room. Make a few steps forward upon your arrival and listen to what Harley Quinn has to say. Start off by turning left and hacking into a control panel (using a sequencer) located close to the window.