Repairing bedroom door jamb
Looks to me like the best fix is going to involve replacing the broken door jam and trim. It's fortunate that the woodwork is painted. That will save you having to match wood type, stain color and sheen of clear finish. You may be able to use a wider piece of finish trim to cover the chipped paint on the inside (knob side) of the door frame.
Do you own any woodworking tools?
Is this a rental? You might be better off being honest with the landlord if that's the case.
"The physician can bury his mistakes, but the architect can only advise his clients to plant vines." -- Frank Lloyd Wright