We had a customer use force to try and open a locked exterior door and broke the frame. The maintenance man did not know how to fix this and took a section of the frame off to repair, and now cannot get it put back together.

Probably just broke the jamb, remove what's left of the casing, use glue on the back side of the damaged piece once it is back in place use some hand drive finish nails 8penny or bigger if possible, sand the crack and fill, prime and paint, you said the piece didn't fit back, there may be a screw or nails that's still in beaded the wood, but if it came off it will go back on.