After wanting to figure out a simple way to fix a shaft that broke off very short, I finally gave up and just replaced the potentiometer. I would have saved a lot of time if I had just done that in the first place. But I do feel your pain and frustration that there ought to be a way to just fix the shaft......
If you still have a bit of old shaft on the pot you can use a bit of brass tubing and some epoxy to join them together. I've also seen short shafts extended this way.