Medical Facility Riddles
In the Secure Access area:
"Are you going to take your hat off to Harley, Bats?" is what you'll hear and you need the Bat Claw for this riddle. Head into the Sanatorium and go to the elevator. You'll see a vent on the wall, and you need to use the Bat Claw to pull it down. Then, get in, head down the shaft to the Morgue. You'll find another vent at the bottom of the elevator shaft, head in it and follow it until you drop into Harley's room. Then, just zoom in on her hat on the desk.
In the Surgery Room:
"Shhsshh! Rumors persist that Tommy Elliot operates in Arkham. Can it be true?" is your next riddle. Get on the balcony upstairs (where those mobs of henchmen dropped down from), and you'll find a large memo board on the wall. On the board you'll see the surgeon's roster. Zoom in on the roster and you'll find Tommy Elliot's name to solve the riddle.
In the Experimental Chamber:
"Is this bear the Bane of his life?", while not much of a riddle, it's the next one you'll get. When you go through this area, you will see Bane's teddy bear on your left side, by the Riddler's map.
In the Patient Observation area:
"Was this fire fly too hot off the press?" is your next riddle to solve. When you have gotten all of the gas out of this area, you need to find the lockers, which are in a smaller room. In one of these lockers, you'll find some newspaper clippings stuck to one of them. Find the one that talks about the Firefly and zoom in on it.
In the Sanatorium:
"A question can only be answered from a new perspective. Don't you agree?", for this crazy riddle you will need more Explosive Gel. When you get into this room, you'll find a part of the floor on the left side that you can break. Use your Explosive Gel to bust it and then stand by the side of the hole and turn on your Detective mode. You'll find the big part of the "?" on the side of the hole itself and the dot is on the bottom.
There is also another riddle here, and you'll hear this: "TICK! TOCK! News flash! Someone is not getting out of here alive!". When you are in the area where you have to get Dr. Young away from the mobs that the Joker sends, you just need to find the cell that is on the lower part that has skulls all over it.
In the Morgue:
"What silent killer of the oceans can be found in a tiny jar?" In the Morgue, you will find a small jar on the end of one of the tables that has body parts that were recovered from a Great White Shark. Just zoom in on the jar, and check the label.
In the Medical Foyer:
"Is the generosity of our benefactors on the Wayne?" is a great little riddle for your next area. When you're in the foyer, right across from the front doors, you'll see a gold plaque that says the medical area of the hospital was funded by Bruce Wayne Foundation for the Criminally Insane. Focus on that and you'll solve this last riddle for this area.