Stay away from extractors if at all possible. If you must use one, be sure to use a T-handle to turn it. Never use a regular wrench or crescent wrench on a bolt extractor. The uneven force applied that way will snap it off for sure, and then you are hosed, you will never drill it out by hand. If there is enough sticking out to weld a nut to, that is okay to try, but if it doesn't work you can end up inadvertently heat treating the bolt on the surface, making it much harder to drill. The best way in this situation is to get a well centered smaller hole started (1/4 or less) and drill through. Soak it good in PB Blaster or Kroil or a 50/50 mixture of trans fluid and acetone. Then drill in 1 or 2 steps with larger drills (preferably left-handed) up to the thread root diameter. After that you can pick the remaining metal out of the threads with a scribe. If your hole starts to get off-center, all is not lost. Just drill with the largest drill you can that will get to the thread root on one side and you can use a center punch to knock out what remains of the bolt. Be sure to tap the threads when you're finished, but be careful because remaining metal can jam in the tap flute and break it. Go slow (1/4 turn at a time) and reverse the tap frequently to break up the chips.
I have been removing broke bolts for 27 years. We rebuild turbochargers and broke bolts are the norm in that business. The worst ones though are the bolts in exhaust manifolds or in the heads because you can't get them on a machine and have to do them with hand tools like your situation. I once had a customer bring me a Harley-Davidson head with a broke exhaust stud that he tried to get out himself. It had an ez-out and two drill bits broke off in the hole. It took two hours working with solid carbide drills and end mills to fix up that mess. It cost the customer $120 when it all was done. If he had brought it to me to start with I could have got it out in fifteen minutes and would have cost only $15.