if you do drill use a left handed drill bit or whatever they're "technically" called. Turns counterclockwise. I dunno if it will work on a head bolt, buit sometimes it will catch and back the bolt out.
I'm assuming you are talking about a manifold bolt and not a head bolt. If you can access the hole with a drill while it's in the car, buy a smaller titanium drill bit or other VERY hard bit and drill the center of the broken bolt. Then insert an easy out and back the bolt stub out of the hole. I've done this with water pump bolts and have been able to save the thread. If the threads get buggered up try chasing them with a tap. Usually you'll have enough thread to align the tap in the hole. After you have tapped the hole see if a bolt will run and tighten in the hole. If not you could drill the hole oversized and insert a helicoil(sp?). The kits are expensive but they work.
centering your pilot hole can be a hassle. Usually the bolt breaks of without leaving a flat surface. The drill bit will want to "walk" on the surface to follow the contour. To avoid this, or at least minimize it, Take a centerpunch and center the punch exactly and give it a heft whack with a big hammer. Might take several blows to put a dimple in the surface. Initally, run your bit at a very very slow speed to get it to "bite" into the dimple. You might have better luck with squeezing the drill trigger off and on, like rev'ing a motor. Once the bit catches and has some depth, you can speed it up. Too fast and it doesn't drill - get the right speed and keep the drill bit end cool with WD40 while drilling. Yes, if you can use a drill press, great, since it will keep the bit perpendicular to the bolt hole. When you chase the thread after removing the bolt, use a thread chaser rather than a tap. A thread chaser is shaped differently at the end so that it cuts the threads at the bottom of the bolt hole. A tap is tapered and doesn't cut the threads at the end of a blind bolt hole. Oiling the thread chaser with WD40 or sewing machine oil helps. Also, if you use a split point drill bit, it will be less likely to "walk" when starting the hole. Drill the first hole with a small (1/8th) drill bit and then drill once more with a bit large enough to get the ezy out into the hole. I agree with the idea of drilling with a left hand thread bit for the second hole. Good luck.