You can finally climb up to a new platform. Head forward and it shouldn't take too long for you to find a new searchlight. Pressing the action key will allow you to end this scene and Batman will be returned to the Main Hall. Notice that you're standing inside a small, dark room. You must throw a Batarang at a nearby rope.
You can now jump over the railing and glide for a few seconds until you've reached the ground level. Open a door located near your position (screenshot #1) which will allow you to enter the area marked as the East Wing Corridor. Be careful, because an enemy with a stun gun is waiting for you just around the corner. There are two ways of defeating him. You can make a jump TOWARDS him, allowing Batman to stun him while he's in the air, or you can grab him with your Batclaw. In any case avoid getting stunned, because you would lose health.
Before you leave this area you might also want to visit the bathrooms, because you'll encounter a new thug there. Perform a silent takedown on him before moving on. Once you're ready proceed to a new door leading to the Warden's Office. Watch an interesting cut-scene upon your arrival to find out that Zsasz is holding doctor Young hostage.
Make sure to follow my instructions, because otherwise doctor Young may end up dead and you'll have to replay the entire scene. Press the crouch key and carefully approach the room where the hostage is being held. Press a corresponding key to order Batman to take cover behind the right wall (corner cover).
You must now choose a Batarang from your inventory. Aim the Batarang at Zsasz. Don't throw it just yet, because the inmate is hiding behind doctor Young. Instead wait for Zsasz to turn his head and use this opportunity to hit him with your gadget. Witness a new cut-scene after which you'll be forced to take on a large group of enemies.
You should focus on eliminating the inmate with a stun gun for obvious reasons. Once he's been neutralized take care of the rest. Activate detective mode when it's safe to explore. Your objective will be to scan a small object found on the ground. Doing this will initiate a new investigation, allowing you to continue with your main mission.
The tracks will lead you outside, so you'll have to return to the East Wing Corridor. Proceed to the opposite end of the hall and find an entrance leading to the Main Hall. Make sure to grapple up to one of the gargoyles as soon as possible, because enemies equipped with assault rifles have arrived in the area.
I wouldn't recommend raising the alarm prematurely, because the enemies would stand close to each other, making it next to impossible to surprise them. Instead use the gargoyles to get closer to the entrance. Drop down quietly and perform a silent takedown on an enemy standing close to your position. Once he's down return to the gargoyles before his collar sounds an alarm.
There are two other inmates for you to take care of. It would be a good idea to drop down to a metal balcony and to perform a Glide Kick from there (screenshot #1). Once you've knocked out the first guard quickly punch his colleague. Don't forget to finish off both enemies before they have a chance to stand up and fight back. You may return to Mansion Entrance Hall and from there to Arkham East.