You said that your could touch the far end of the broken bolt, Correct? If you can get a drill in there drill until the new hole hits the broken bit then get the largest bit that will fit in the hole and again drill until you hit the broken bit. With a straight punch the same or slightly smaller diameter than the broke bit, and a good size hammer, drive the broken bit out.
If you cannot drill from the bottom get a 3/8 carbide 4 flute end mill center cutting, and the most powerful drill motor and that mill cut that drill bit out. You need lots of pressure to do this by hand, be careful not to cock the drill motor, carbide will break.
Don't bother with a carbide drill then can only be used in a rigid setup DP or Mill. If the work can be put oor a Mag drill used, then a spade carbide drill will cut that bit out in quick time.