Find two crosses in the library. Take one for the death chalice by the first door. Then take the other for the death chalice by the locked (silver) door at the upper end of the corridor.

Enter the dining room by any of this room's doors and neutralize the four ghosts in it with your axe. Then take the silver key and the health chalice in this room.

Enter this corridor. In this section of the floor you will encounter several armours and ghosts. Try to save your plasma ammo as much as possible and switch between the plasma gun and the axe. You can shoot the armours in the corridor in any order you want.

Make it to this washing room for the brass key. Find additional health chalices in the purple bathroom next to it.

Continue.