OK, so you've identified the problems clearly enough.
It sounds like kit has gone out (been taken out) of the Library without any record of who borrowed it. A fundamental breakdown of procedure, then. Not good.

So, what about the solutions?
An access key pad for out-of-hours collections?
A registered "swipe-card" (to allow access to the Library) issued to each ward and/or trusted staff member?
Cameras in the library?
I'm sure there could be a whole number of ways of going about it, depending really upon the details of the set-up where you are.
But "far from the clinical areas" sounds like a difficult one to address.
How far, I wonder? It sounds like the only answer there is the obvious one ... move the Equipment Library to a move suitable location!
Meanwhile, how do you generally operate? During the day time, do you deliver and collect the kit, or do you have the staff come to the Library to collect? If so, that could be yet another bone of contention.
At many of the Libraries I have seen, the
Librarian spends a fair amount of time each day "out on patrol", as it were. That is, not only delivering the kit, but also dragging it back off the wards!
