If you were tapping stainless, did you back the tap-off after a couple of turns, then broke the tap? Many stainless steels work-harden to a very hard level if you back the tap off. It is best to use either a gun-tap (forces chips forward) or a spiral flute tap. With these two styles you don't need to (nor is it advised) to back off the taps while cutting. Spiral flutes are needed for blind holes.
If the broken tap is not high-speed steel, then you should be able to break it up into small pieces. If it is high-speed, then you should be able to cut-it out with carbide tools in your dremel. With stainless, it is nearly impossible to use tap extractors or to force the tap around with a punch.
If the carbide tool, or breaking it up does not work, then you will need to A) scrap the part, B) find someone with an EDM (as suggested above) or C) move the hole to a new location if possible.
Dave.