Carbide tipped drill bits will cut through broken high speed steel drill bits and broken EZ-Outs like butter. They should make short work of that. Use slow speed. You want one that looks like a normal twist drill, but has a little chip of carbide brazed to the tip like this:



https://www.amazon.com/Jobber-Carbide-Tipped-Standard-50359/dp/B00PM4T9QU