I once had a friendly dentist that removed a broken tap in his office. Best way I've found. A dremel tool with a diamond tip will take it out, I learned from the dentist. Brownells sells a tool with fingers that go in the tap flutes and if you are lucky you can turn it out. If you have the right size punch you may be able to break it out. Find a sharp corner and chip away. Carefully, of course.
