I have broken several taps and usally have to take a punch and drive the tap all of the way through and then redrill. these have saved me a lot of work removing broken taps. they slide into the flutes of the tap and then use a wrench to twist the tap out. very handy.
wow those are a great idea it is a pain in the a$$ to get broken taps out if they break flush with you piece if they dont break flush vise grips seem to work good