Broken bit nightmare

My buddy & I were installing a RE SYE hack n tap on my Cherokee without removing the TC. Everything was going smoothly until he snapped off a bit while drilling a pilot hole in the hacked end of the TC output shaft. It is a 1/16" titanium bit drilled in about 1" deep. The bit broke not only at the end of the hacked shaft but also about a 1/4" into the shaft. (So there is roughly 3/4 of an inch still in there.) Though it probably wasn't the best idea, we grabbed a slightly larger titanium bit and drilled in until we hit the broken bit and couldn't go any further. If you are familiar with RE's hack n tap system then you will know that there isn't much tolerance for screwing up this hole which will ultimately need to be about 5/16 of an inch when all is said and done. But in the meantime, we are stuck. Can anyone recommend a way to either extract or drill through the 1/16" titanium bit without causing any unnecessary damage to the output shaft?? Any advice would be appreciated.

Thanks

PM sent to you for your problem...

Just in case anyone in a similar situation stumbles across this in a search, the answer to my problem turned out to be a tungsten carbide cutting bit for a Dremel. It took a couple hours to get through the stuck titanium bit, but that was far quicker than any of the progress I was making with anything else I tried. One bit will do the trick. Cost me about $8 bucks at the hardware store.

Thanks again to AlohaBra for pointing me in the right direction.