What kind of taps are you using? From the pictures it would appear they are a spiral point tap, which are going to push the cut chips ahead of the tap.
I would suggest looking for a spiral fluted tap which will push the chips out of the top of the hole (and are a it bit more robust from my experience in harder material), which may keep them from breaking. Also, what thread percentage are you drilling for?
I've been in the machining sector for 10+ years, but as with all advise, it's worth what you have paid for it.