Because the locking nuts have a rotating guard around them, neither of the extractors would remove them.
Following several days of cold weather we caught a warm spell, which I took advantage of. Before attempting to weld the ground cable was attached directly to the wheel to be worked on. This would eliminate any potential of weld current damage to that wheel bearing. My usual method of laying a nut over a broken stud did not work due to gassing. Stitch welding in open air allowed the contaminated material to blow off and created a nice base over the end of the locking lug nut.An 1-1/8" nut was chosen to give optimum clearance for mig welding it to the lock nut. The ground cable was moved to the other 3 wheels and the process was repeated and allowed to cool for about 20 minutes each. This method turned out to be simple and cheap. 4 new nonlocking nuts replaced the locking nuts, with an added bonus of being able to do tire changes without needing to hunt down the key.

