If its still in there,
My two bobs worth.
Blow the swarf and other crap out of the flutes with compressed air @ 10 bar. Then drill down to the bit with a bigger drill to open up the hole a bit if its only 10 mm down and then get some sort of two pronged thing, as previously suggested and twist ,wobble or tug on the bit.The drill bit is only stuck on the swarf being cut at the tip and will only need a small amount of movement to loosen it. If friction is a problem use some wd40 type stuff. What size is the actual drill bit? If its to drill out a 6mm stud,it would probably be 4mm maybe 5 mm or 3.2 mm bit. Can you drift down and give it a sharp smack with a hammer to push it through the final grippy bit if it has almost broken through?
For the two prongs,I have used small circlip pliers to remove broken taps and drill bits in the past. If its an easyout down there, chuck out the head
Steal an idea from the two banger trailies and fit springs to the head and then to the header, till you can get it fixed.