I know it's too late now, but the best solution is to not break the tap in the first place. Here's what I've been picking up as I go along:

Hole size can make a lot of difference in how easily a hole will tap. Used to be that I drilled the hole just a bit under-sized, in the belief that I would have full threads that would be stronger. As you might imagine, this resulted in hard threading holes and more than enough broken taps. Turns out that this wasn't true and I was actually ending up with weaker threads as a result of my efforts. An under-sized hole will result in the tap needing to bore the hole to size and actually tearing the tops of the threads off, rough threads, and an increased chance of a broken tap. Drilling the hole just a bit oversized will weaken the threads very, very little and make for a much easier tap and clean sharp threads. If you have a set of number & letter drills, it's not much of a chore to pick a drill that will be 2 to 3 thousandths over-size. Use that and see if it doesn't make a huge difference.

Next thing is; it's not a sin to back up with the tap and clear the chips when you feel them binding. Shop vac comes in handy when working into a blind hole and can also vac off any that are stuck to the tap.

Most broken taps are the result of actually breaking the tap by bending it and not by twisting it off. Starting the tap straight in the hole is also important. Tapping guides are an excellent tool. Bench mounted ones sell for less than $80 and work well. For small taps that are inaccessable otherwise, I found this little gem for about $10.

 They sell it here but looks like the price has crept up a couple bucks. http://www.shars.com/products/view/7547/SelfAligning_Tap_amp_Reamer_Holder

Next issue is what to use for tapping lube. I've tried a wide variety of stuff, from motor oil, synthetic motor oil, hydraulic oil, slick50, WD40, high $$ tapping fluid, #2 diesel, pipe cutting oil and have never really been all that impressed with any of them. So, I've kept reading old machining manuals and they keep talking about using lard for a lube. More modern guys say Crisco works as well when turning a nice finish on stuff. Surely it couldn't be that simple, could it? I asked my bride to pick up a pound tub of the stuff when she was at the grocery store. Apply a bit with a brush to the tap or whatever you happen to be turning that you want a nice finish on. It makes for very easy turning taps and a nice clean thread.