not knowing exactly how the drill is broken, and what material the bit is broken off into, limits my response to mere conjecture, but here it goes...
if you can open the top of the hole with a slightly larger drill or endmill- down to the depth of the broken bit, you may be able drill, invert the part, and tap on the backside.
the broken bit may fall out, or at least be more accessible for a pick or left hand drill to extract the broken section.
you could then possibly tap the hole to accept a piece of allthread and re-drill the hole back to the original spec size, or possibly simply retap the hole and use a larger fastener for that bore.