The bit that broke in the hole would be hardened so you wouldn't have been able to drill it out with a normal bit anyway, best way I have found if that happens & you don't have access to specialist equipment is.... If you can drill a small hole on the other side in the same place, take a bit of care when you get close to the end of the drill bit or you could end up with another broken bit on the other side lol.
Once you have drilled a hole use a bit of hardened rod to try I tap the bit back out. Does work if you are careful but a bit of mucking about is required so if the hole is not that important then normally best just to drill another one lol.