Did you get Raan's blessing? (in the northern room on the west edge of the level)
You shouldn't need to do anything else to get the key from the first instructor (in the south west corner room). That key opens a door in the hallway they leads to the library, rather than the library itself.
Which door are you trying to open? IIRC, from the hallway you should just be able to walk into the first room of the library, and then you need to talk to the librarian to get access to the other room.
