There are several ways to remove a broken off bolt. The ways I most often see are:
Use what's called an "easy out". You drill a hole down the middle of the broken bolt, then screw in a screw that is reverse threaded, eventually popping out the bolt. This works, and is pretty cheap, but can be difficult depending on the broken bolt.
Another method is to weld a new bolt to the broken head of the old one.
A good machine shop should be able to do this for you.