It is not uncommon to bust your taps when threading. Unfortunately, there is no one size fits all solution for extracting a tap since many variables are at play. The metal type combined with how exactly the tap became stuck will determine which method you should use to try to remove it. For example using a tap extractor on cast iron is a completely different story than using it on steel. Because this process is so situational, if you find yourself in this situation it will be helpful to grab advice from someone with similar experience. No better place to ask around than the Practical Machinist forum.
But one thing is for sure, and that’s that patience is required. If you rush the process or pursue a makeshift option, chances are you will worsen the problem. There are many cases where a machinist will be overconfident and go at the tap without reading instructions or researching how to pull off the meticulous act. Then they will write off the tool they used like it is the tool’s fault. Whereas the machinist who finds patience and is thorough, could remove that same tap with success.
If you want to avoid this headache, it may be helpful to know WHY taps commonly break. READ ON at PracticalMachinist.com.