How do you remove broken alloy nuts from an x type jag?

I had the same problem, and had luck with the following process.

first I tried taking a smaller socket and beating it onto the remaining section of the lug nuts and using an impact gun to try and remove it. (had 2 successful lug nut removals this way)

If that didnt work, I used a small drill bit and started drilling pilot holes into each lug nut, being careful not to foul the threads on the lug stud or to drill all the way through into the wheel. Then after the pilot holes were done, i used a larger bit to create a larger hole for the head of a punch to fit into. Using a hammer and punch, I tried forcing the nuts to spin off by angling the punch to make them spin counter clockwise if they were going to move at all! (had about 5 removed this way)

If you still don't have them off, then try continuing with the drilling, make your larger drill bit cut deep into the lug nuts, and again try not to foul up your threads on the studs! When they are almost all the way through, stop and use a cold steel chisel to try and split the lug nut. This may take several times until you weeken the lug nut enough to "peel" it away from the stud without damaging anything.

If you damage a stud in this process, you are either going to replace the stud, or clean it up with a tap and die set if its only slightly damaged.

Good luck!