"Probably a key to a door in the hospital. It's covered in rust."
— Item description.
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.
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 it. For online play, the location for VERY HARD changes to the Intensive care unit where it is stuck between the plant boss and the corner wall (you'll still need the lighter to see it).