My experience drilling out broken studs and bolts has been absolutely horrific. I was dreading this task because the last time I failed, I had to replace the cylinder head.
I had an exhaust leak at the passenger side manifold that was causing the car to sound like a go-kart when it accelerated. I tried to tighten the nuts on the studs and discovered they were broken. I went ahead and removed the manifold and discovered a total of three broken studs out of six.
One of the studs was broken off above the hole and I was able to grab it with some vice-grips and twist it out.
The other two were broken below the hole and would require drilling to remove.
The angle was horrible and there was very little clearance to get a drill in there (even with a right-angle attachment).
I ordered a special tool that helped assist me in centering the hole and drilling straight.
It has a threaded insert that is the exact same size as the stud. There is a drill bit right down the center. Since the stud was broken below the surface, I could use this guide to grab one or two threads and position the drill directly center and at the correct angle. This made the task so much easier.
After drilling the hole, I used a screw extractor to twist out the remainder of the stud.
One other thing I learned through this process. Cobalt drill bits cut through steel like butter. I finally broke down and dropped a bunch of money on a nice drill bit set. What a difference! This also made the job significantly easier.