Pneumatic 1/8" die grinder and a diamond burr does it for me. I've done 4-40s in the past, takes a steady hand and good lighting. You cut the center out of the tap from the side(s) and pick the pieces out. I've never, ever, had good luck with any sort of tap extractor, the taps are usually siezed in the material and those itty-bitty fingers have absolutely no chance to back the thing out without snapping across. Since I've gone to using a tapping jig with bushings, I don't break taps, large or small, anymore. I can't understand breaking taps in cast iron, though, that stuff is usually one of most free-cutting materials there is. You might have hit a chilled spot or aren't cleaning the tap out enough. One way to do blind holes is to fill them with cutting wax, I use Castrol brand, then tap them. The extruding wax pushes the cuttings out of the hole. The wax can be cleaned out with a blast of brake cleaner, if needed.
Stan