Darly, if you have the masonry bit and a diamond wheel to sharpen it, give it a try.
Can't remember who makes them, but there are carbide bits made for drilling out broken taps. The ones we use, the section where the flutes would be is a somewhat square cross section, with a reverse taper...bigger at the tip. Opposing flats form the point. There are no positive cutting edges...pressure and friction will "melt" the metal being drilled. These work on occasion...require a LOT of down pressure.
A slower method would be copper tubing used as a drill with coarse lapping compound. It may take 1/2 hour to do one hole, but it is cheap, safe, and almost guaranteed.