How to remove broken hinge which is stuck inside
That style of hinge is generally made from a pair of metal cups and a piece of round bar.

It definitely looks like a good candidate for heat. There will be damage done to the surrounding paint. Either warm the top piece with a torch, or cut the whole thing off and weld in a new hinge. It looks as if there's welding to be done there anyway to re-attach the hinge to the door.

If a torch isn't available but a welder is, and if you'd really rather not cut off the lower piece of hinge, then you could weld a couple beads up the exterior of the top portion and grind them off later. As an added bonus such beads might give better grip to the locking pliers you'd want to use to grab and twist the hot hinge.