What kind of tap? Greenfield, or a four dollar hobby tap?
Drill the hole all the way through, and you stand a chance of driving the broken part out the bottom.
Also, a #55 drill will give you plenty of threads, yet allow easier tapping.
Ten years ago, a good tap was eight bucks. Mine break after several hundred holes, and I buy them three at a time. Have no idea what a good tap costs now.