I have never replaced a heel, I just darn them, but I did replace the palm on a pair of mittens. The palms were so bad, they were beyond darning. They were a beautiful pair of mittens made with an old sweater with a fleece lining.
I made a whole new palm starting with about 5 stitches with heavy yarn on small needles. Made a nice heavy palm. I increased the 5 stitches each row until it was as wide as the mitten. I knitted down to the base of the thumb and then cast off about 2/3 of the stitches and continued up the thumb until it was long enough. The end was decreased and knit together at the tip of the thumb. Then I hand sewed it on. When I made the second one, I had to put the thumb extension on the other side. They looked really cool. My granddaughter and her mother really like them. The granddaughter is 16 years old.
So you could cut out the heel, probably sew the cut edges at the sides, pick up stitches where the sock is still good, and knit until you come to the good part. There you can weave the stitches together. If you cast off and sew these ends together, there would be a lump which you wouldn't like in your shoe. I hope you know how to weave stitches.