Originally Posted by Doug64

As Staehpj said, use a center punch to get a good stable place to start the drill bit. also use a smaller drill bit (3/32" or even a 1/16") to make a shallow hole (<1/8") providing better guide for a 1/8" (3mm) bit. Then a 5/32" (4 mm) This will give you a " buffer between the drill bit and the threads. Let the drill do the cutting; don't put a lot of pressure on it. Once you have a small hole through the bolt it is easier to use some finesse enlarging the hole as you use the larger bits. Usually the drilling is enough to loosen it up. The real challenge is finding a center punch that will fit down a 5 mm hole without messing up the threads.

I'm assuming that the bolt broke because it was frozen in the mount or you over tightened it. It might help if to put some penetrating oil in the braze-on hole and let it set for awhile before working on it.