I haven't had the need to attempt this project, so I don't know exactly what you're dealing with, but...you have any friends in the medical field? I'm a veterinarian, so I've always had available lots of these:
Don't know how 'stuck' your bit is. If it's so jammed you'd need the force of pliers to get it out, a hemostat probably won't do the trick. But if it's just a matter of reaching and grabbing it, these things are great.