Proceed left until you reach dark-brown door embellished with some metal ornaments (left from the squatting patient). While you are zooming in you must open your inventory and use the key you just got from the doctor. The entrance key is rather unusual - nevertheless, use it on the lock on the left. Then, in another zoom, click the gear icon on the active spot in the middle of the key. A puzzle will appear.
Match the key to the key hole
You have to manipulate the wrapped fragments of the key so they match. You do this by dragging the endings of the fragments into the matching structures of the lock. You can see below how the key should look.
When it turns out that even now, the lock can't be opened, open the inventory after speaking to the doctor, move the key to central slot and examine it up close (the '3' key). By dragging the mouse from right to the left you can rotate the key - you want to see the hole in its bottom part. Examine the bottom of the key and the hole, which is a bit higher, and Kate will ascertain that the key is incomplete.
(If you want to get Friends for life achievement then you should go back to Anton and Leon, the two oldest patients, that are nodding off next to chess behind the blue ornate door. Speak with them about the key that is not matching the door).
Exit the zoom and go to the Doctor's office. Check out his desk (interactive spot on the chair) that's standing on the right of the polygraph. When you zoom in on the drawer, click the items on the right and drag to the left, until you will spot the brochure with a red building on its cover. Take it and skim it through - you will see a drawing of the key. Click the interactive spot in the top of the key, and when the equipment appears, interact the key with the drawing from the book.