Blind or through hole, I've always resorted to chipping or grinding out broken bits. Sometimes you can take a small centre punch or awl and tap on the drill bit in its reverse direction and it will start to reverse itself enough that you can use a dental pick to work it out.
Yeah, did jobs like this for a living for a while, got pretty good at it.
Scriber, dental pick dental burrs and an air drill, carbide mill cutters and a milling machine, etc. Be very aware that breaking off carbide anything in the same hole as the already broken item, is going to make your life really miserable... Have a plan for what you are going to do if it all goes haywire!
Never had an EDM to use, so never went there. A friend of mine built a simple EDM burner out of a 12v battery charger, some copper house wire, and some pieces of plastic pipe, which worked pretty darn well, considering. Essentially a solenoid that used the electrode to make the current path, which then retracted the solenoid, breaking that path, causing a spark. Gravity drops the electrode down again, the process repeats, makes a hole, eventually.
Best of luck, stay patient, and be persistent. The drill would have broken because it wedged on something, often you can poke and prod at it until it un-wedges. Otherwise you will have to look at more violent methods.