I had 3 head bolts break because they were corroded to the thread from general conditions near an engine. They also had corrosion near the head, attaching them pretty firmly to the head. With lots of hammering, penetrating oil, a torch, and some wax, the head was broken (not literally) free and removed. All 3 bolts were broken above the thread by a fair amount, one broke free with a vice grip and more heat and hammering very easily. The other two were near impossible to remove. They ended up breaking again flush with the block. With quiet contemplation, we decided to use a magnetic drill press to remove said bolts, starting small and moving up, then using a tap to clean out the remaining metal.
It was not pretty at first, and a little scary with a drill on the block, but it can be done. I would suggest doing this as a LAST resort. Broken bolt removers of all sorts did not work and ended as a waste of money in all cases on these bolts also.
These three small pieces of metal turned a valve job into a complete nightmare!