A large
protractor, or is there a device that fits inside interior angles (such
as inside the door frame) to measure them?

I have a device, not sure what it is called, it has a wooden handle
with a thin metal blade that folds out like a pocket knife that is
handy for finding angles, it is a pretty common tool, not hard to find.
You may encounter another problem, once you find the angles, the cuts
may not have the same face length. Unless it is horribly out of square,
it may be better to just cut the miters 45 and at least have part of it
right.