Ryan, Ron has a very good question. If you found them broken and knew they were there before, they may not be all that stuck. Still there's a common denominator, which is they need to be drilled. Ron's point adds to the possibility that several of us have mentioned. Namely, that a Left-Handed Drill may just bring them out without further heroics.
You deserve a compliment: You're taking the time to plan a strategy, not just grabbing the drill motor with any bit and going for it. Continue to take your time and get it right, even if it's a few days to get tools. Then, once you have a hole through to the other end of the broken pieces, you can do some meaningful soaking with penetrant. Drill them, soak them, beat them, maybe an Extractor. From the way this discussion is going, if you've drilled small holes and the pieces do NOT back out with the Left-Handed bit, hold off on the larger size till you've done a couple cycles of soaking and tapping it with the punch, before you drill. The bigger bit may find the pieces easier to spin than to drill. In reverse, that's what you want!
