Plumbing Problem - had dripping faucet in bathtub. We were attempting to remove the screw that holds the handle in place, but srew was defective, so we drilled into it, reversed the drill to get the bit out, then used a screw extractor to remove the screw. The screw extractor broke off! (I DO hope I've made my problem clear!)
What do we do from here? If only there were drill bits that would drill a CIRCLE into and AROUND the little screw in which the screw extractor is embedded, we'd have a solution! But, lo, there doesn't seem to be such an item.
Any assistance would be SO appreciated!
Thank y'all SOOO much!

Buy a small expensive bit smaller than the extractor, drill a hole directly in the center of the extractor then drill a little larger hole and then a larger hole till you get a hole that you can tap and put in a new brass screw. The bits I am talking about in that size average about four dollars each,
Second option take the stem to a good plumbing supply shop and buy a new stem and screw.

Even with the best bits, drilling a hole in a hardened screw extractor is not easy. It may not even be necessary. If the head of the screw is gone, the handle should pull off. Once you have the handle off, on all but the oldest faucets, you will be buying a new cartridge complete with a new stem. If it doesn't come with a screw, you can buy a screw.

Depending how special the handle is, even if there is too much of the screw head left, you can cut or break the handle off and buy a new one of them too.