If you work on old stuff this happens, patience is what it takes to be successful with this issue. The following is what has worked for me, spray some penetrant and let sit then center punch make sure to be in the center of broken bolt. Start with a small drill bit and drill all the way through the bolt, once through you'll see how off center you are and that's ok it just limits the next size or the final size hole you can drill with out damaging the threads. Drill removing ass much of the bolt by drilling out with larger drill bit but staying away from the threads after use an easy out with success, like I said be patient