I broke a carbon steel M8 tap in mild steel, and tried various ways to get it out. Eventually I put the workpiece on an anvil and hit the tap with a masonry nail, using a hammer, hard. Reground the end of the nail a couple of times to fit cracks etc, but ..
This caused the tap bits to fragment, and I was able to remove the fragments and even retap the hole with another same-size tap.
Yeaaay! :)
(btw, the other tap was a top quality expensive English HSS tap - and wow, that's all I can say. Two fingers to turn it .. you don't usually get what you don't pay for, but you do get what you do pay for, at least sometimes)
-- Peter Fairbrother