A neighbour introduced me to the use of stainless rods for jobs like this. He uses it on bolts broken off even deep below the surface of what ever they're in - the rod flux will keep the weld off the threads if you don't get too close.
He gets the puddle very hot and builds it up to where he can weld a nut on. Then tries to rock it, using bulk WD 40 until he feels a click as it starts to move.
Needs practice though. So far I've used it successfully on those broken near the surface - not up to the deep ones yet.