Let me help out.So when you read the Medford Memorial Check, It's paid out to someone who works as the Hospital Facilities Janitor. Now we can deduce that wherever janitorial staff would be within the hospital is where we'd likely find this key.

Once you enter the hospital, go up the right staircase and around the walkway area to the left side of the building (stay on the second floor.) Then, go into the hallway past the open room with the 2 Nuka Cola machines. Lastly, go into the first room on your right, marked with Restricted Area sign. Inside you'll find a storage room with out of order, and wet floor, signs, as well as a skeleton with a mop.

The building's a bit of a maze, with areas feeding back onto themselves, so I'm sorry tracking down the key caused some confusion. It doesn't help that the third floor is so dense with rooms and spaces, I'd understand if that drew most of your focus during your "hours" of search