"Probably a key to a door in the hospital. It's covered in rust."
The key is found in two different places;
HARD; Room 202, on the bed in front of the door.
VERY HARD; Main Building 2F Hall, hidden behind the broken elevator doors; it requires the Lighter to see.
At first, the description reads that it's probably a key to a door in the hospital. The real purpose of it is to remove the cover covering the button in the Administrator's Office, which only appears on the said difficulties.