I think the door was locked when the knob fell off (or was taken off by someone trying to help?).
I've tried pulling and turning various bits with decent force to no avail, but I wonder if I'm missing some technique.
The part of the door knob I've got doesn't have the spindle. That's stuck inside the room.
I've received updated information from the party at fault that the door was slammed into and, despite looking perfectly fine at first glance, the latch bolt seems to have broken or dented inside/around the face plate / door frame.
I was prepared to cut the thing up and replace it, but after trying using a lot more aggressive force knowing the mechanism is shot, I was able to pull the latch bolt through. It had been bent / dented and wasn't going through the face plate without lots of encouragement.
The piece you are looking at is an outer tube containing a sliding piece that retracts the latch. Normally the knob has a piece of metal with a C-shaped cross section which extends through the opening, and when rotated either the top or bottom of that C comes in contact with the inner piece and retracts it.
So you could just insert a spare knob, turn it to retract the latch, and open the door. Which is usually the easiest solution.
Or you can do it the harder way by using a tool to push the inner piece away from the edge of the door (to the left, in your photo), manually retracting the latch.
You should be able to slide that metal part to the left to release the catch.
If the door is tight, then a gentle push on the door may take the pressure off the tang.
I find a screwdriver or sturdy rod usually works.
Combining the screwdriver with a block to give a pivot can also help.
Inside that center hole is a black slide that is operated by a cam on the knob shaft. The cam pushes the slide to the left, pulling the latch pin out of the frame. You need to find some way to push the slide to the left.
