lohengrin,
You were probably using a HSS drill bill and were drilling too fast, not removing it often enough to cool and to clear the channel, nand it snapped from the undue heat stress. Cooled off now, the wood of the shank has tightened up and likely has that bit locked in a vise-like hold.
Not in a million years will you ever drill it out. Drill bits wander seeking the path of least resistance, even the best of them, and I don't care what kind you try, the bit will just seek out the softer wood when you try inserting a new one. You will never drill that first one out.
Super glue? First off, cyanoacrylate depends on a good tight fit. Adding more glue weakens the bond. The only way you have a chance of pulling it out (a 1 in a 100 chance) is to dab a tiny amount on the remainder of the original bit and insert it down then by feel twist it until you feel the parts join up. Use the slow set kind. If by some small miracle you happen to join the parts perfectly and the bit isn't jammed in there super tight, there is a CHANCE it might work. My hat is totally off to you if you succeed in doing this though. Most likely, you will just leave a drop of super glue down in there, and if you otherwise fix the pipe and it remains, super glue has a HORRIBLE taste and odor. It is made in part out of cyanide. Not what I want to smoke through.
If you don't mind defacing the pipe, you could drill through the bowl from the front until you hit the bit then try to punch it back out. Then all you would have to do is figure out a way to plug that hole in some cosmetically agreeable way.
You best option is what derfatdutchman said: leave the bit in there--- get a new, better drill bit, use proper drilling techniques, remove the bit frequently to cool and to remove matter, you might try lubricating the bit with walnut oil; when you hit the old bit, realign until you get the bit to jump past it then keep it on the right path until you clear the bowl, drilling right alongside the old bit. You can probably enlarge the mortise of the pipe to keep its shape round. That old bit just leave in there, it will do no harm.