To make sure you get a good solid ceneterd hole, use a pointed center punch to get a dent in the center of your broken bolt. The drill bit will "walk" on you if you don't. Start your drill in the dent. Start small, like a #30 bit and work up from there. MAKE SURE YOU'RE WEARING SAFETY GLASSES!!! If the bit snaps, and it probably will, the broken end can real easily end up in an eye. A couple bucks for a pair of glasses (go to your local Airgas store, ask for p/n 64051315) is cheap insurance!! Lubricate the bit often and drill slowly, let the bit do the work. If you drill too big, or go offline and drill thru the threads, you can oversize the bolthole, run a GOOD tap thru it and rethread the hole. Use a larger bolt in this case.
THis link has great info.....[
www.asashop.org]
I've done this for years getting broken fasteners out of aircraft. Patience is the key. You might want to run down your Snap-on or MAC truck and get a set of left handed drill bits. They run turning counter clockwise, grab the bolt, and 8 times out of 10 pull it out. Best money I ever spent.
Good luck!!