If you're doing a lot of drilling into metal, spend the money on M35 or M42 Cobalt high speed drill bits. They'll last a long time compared to normal high speed steel drill bits. Expensive but worth the money.
Hmmm well if it's a grade 5 or less that broke off, it shouldn't be too much of a pain to use a regular high speed steel bit to drill it out. But part of the problem maybe that it doesn't drill out right and may need to be retapped.
When ever I drill metal I always use oil so that the bit cuts better. Sometimes I use plenty of it and always take time to stop and let your bit cool down a little. This helps cheap bits drill better and last longer.
I have good bits and I still do this with them. They drill better and last longer.