You have a bench grinder? After you've tapped your hole with the regular tap, take it to your bench grinder and grind a bit of the tip off. You now have a bottoming tap.
When you tap, there are a few tricks... the best is to take a little block of aluminum, or even wood, that you've drilled a hole on a drill press that's just larger than the tap. You can use this as a guide to keep the tap straight to your work piece.
And, when you're actually tapping, don't rush it. Just turn it in in little segments - 1/4 turn at a time. After a few quarter turns, back it out and clean the chips off. Then do it again.