removing a broken tap is no problem at all, providing you use the correct equipment,its something i do on a day to day basis.
in your case you need to use a carbide centre drill, followed by a carbide drill bit, preferably a spade drill, you need to run both of these at quite high speed,
once youdrill a gole through the broken bit of tap, it would be possible to shatter the outer parts of the remaining tap and it should fall from the thread,
the other option which you have is to shatter the tap with a good punch and a big hammer.
relatively easy for someone with basic engineering skills, try your local engineering firms, they'l get it out no problem at very little cost.
ash