A tool I have found useful is a long taper carbide burr.As a field service technician I sometimes have to remove broken screws, field means occasionally under the liftgate of my vehicle in a parking lot, no outlets available, and hand held work.
For removing small broken screws 3mm (occasionally 1.5mm) as shown on the end of the steel rod locktit-ed into an aluminum blind hole I use a AA battery powered Dremel. The Dremel I use was advertised as being useful for cleaning golf equipment. It is fitted with an accessory chuck and the long taper carbide burr is used to start a hole. Being tapered, the hole can be re-centered, deepened, and one side used along the thread root to grind out the remains. A straight grout removal bit can be used on larger screws. A 90 degree bent wire sharpened to a point like a threading tool is used to hand chase the remaining threads and a tap run into the hole.
For removing small broken screws 3mm (occasionally 1.5mm) as shown on the end of the steel rod locktit-ed into an aluminum blind hole I use a AA battery powered Dremel. The Dremel I use was advertised as being useful for cleaning golf equipment. It is fitted with an accessory chuck and the long taper carbide burr is used to start a hole. Being tapered, the hole can be re-centered, deepened, and one side used along the thread root to grind out the remains. A straight grout removal bit can be used on larger screws. A 90 degree bent wire sharpened to a point like a threading tool is used to hand chase the remaining threads and a tap run into the hole.