You won't encounter any enemy units inside the mansion, however it's still going to be difficult to stay live. You can't allow Batman to get closer to the ground, because Ivy's plants have released deadly fumes. Use your Line Launcher to reach a balcony located on the other side of this room. Once you're there drop down to an area below you.
You may open a nearby door to travel to the Main Hall. Use your hook here in order to find your way to a metal balcony located to your right (screenshot #1). Start moving forward and once you've reached some obstacles turn left and use the Line Launcher to travel to an adjacent balcony. Don't forget to aim the device properly, so you won't hit any obstacles and fall.
Proceed to the opposite end of the balcony, jump over the balustrade (screenshot #1) and glide towards the staircase seen in the distance. You'll have to proceed to your right from here. Talk to some of the nearby guards, including Aaron Cash.
Once you've acquired the necessary information you'll have to go back to the first room of the mansion. Return to the stairs and use your hook to grapple up to a metal balcony. Make a few steps forward and this time use the Line Launcher to travel to the second balcony. Drop down somewhere near the door (not too close to the toxins!) and travel to the Mansion Entrance Hall.
You'll have to use your hook again to return to a small roof section located directly above you (screenshot #1). Once you're there choose a Line Launcher from your inventory and aim this gadget at a small platform seen in the distance. You may go back to Arkham East.