Did the bolt break while trying to remove it? If so, it's probably corroded into place meaning you'll wind up using the big guns (heat, like a torch) to get it loose.
If it broke while tightening, the bolt was probably fractured to begin with and that means the broken piece should come out pretty easily. If that's the case, I have in the past used a Dremel and cutoff wheel to make a screwdriver slot in the broken piece and then backed it out. That probably won't work if it's corroded into place.
Either way, the best way to start if you decide against the Dremel slot trick: centerpunch the broken bolt and drill with a LEFT HAND drill bit. A RIGHT HAND (normal) bit will run the broken piece further into the hole. In a lot of cases, the heat and vibration of drilling will make the broken piece back right on out. You might also get a MAPP gas torch and put some heat on it, too. Caution: the O2 cylinders for the little MAPP torches are only good for about 20 minutes, tops.
Keep penetrating oil on it (I have had good luck with PB Blaster). If it was corroded into place, you might as well save a lot of time and keep drilling bigger until the threads are the only thing left, then clear them with a tap. That rarely works out perfectly in the real world, in most cases the drilling is off center enough that only half the threads get drilled out. Now it's Heli Coil time.