Since we live in a disposable age, I don't have a darning egg, which used to be a pretty ubiquitous item found in a sewing basket. I even think my mom has one. They can be an egg-shaped or a concave piece of polished wood at the end of handle. In Britain, I think they call them darning mushrooms. In any case, darning eggs are as scarce as hen's teeth at my house, so I used my glasses case for one.
Start by turning the sock inside-out.
This problem is an entire hole, so it will taking some doing. Run a line of stitches around the hole to stabilize the edges. (By the way, this method is different from the accepted old-fashioned way of doing it, but I think it does a better job of enclosing the fraying edges.)
Then run a 'ladder' back and forth across the hole, making sure you get at least as far as your stabilizing stitches. I usually start in the middle because I think it makes for a more even darn. Then I go across, starting my weave stitches.
Once you have all the warp stitches in, start weaving the weft stitches in.
Once I get the infill done, I do the same thing all over again, only this time, I run the warp and weft stitches diagonally. You won't see them for the most part, because you're running through your new fabric.
And this is the new fabric from the outside. It's not pretty, but it'll hold well enough for slipper socks which is all this pair is good for anymore. I think part of the problem with them is their thickness, which caused the rubbing on Steve's shoe, so I doubt they'd last much longer in his shoes again.
The problem on the other sock was that the hole hadn't quite happened yet.
Working from the inside of the sock again, after anchoring your yarn, you want to do the same kind of laddering stitch, but only do one row at a time. You're working the yarn back and forth, kind of the same way a lace goes into your shoe.
On your back pass, you want to catch the stitch you took on the previous row. This is not a good picture because it doesn't show me catching that lighter green stitch (all my shots of that were out of focus). But you want to catch your previous stitch because you're creating a new fabric over the old.
This is the second fix from the outside. It's a lot neater, so obviously darning socks is easier and tidier if the husband doesn't wait until the heel completely fails before pointing it out to his wife.
There are other ways to darn socks. This is just the way I do it. If I had to darn a pair of hand knit socks, I would probably learn how to Swiss darn, or use a duplicate stitch, or some version of the toe graft or Kitchener stitch, and maybe I'd finally learn it instead of having to refer to the bookmarked YouTube video like I do currently. The Kitchener stitch intimidates me.
The important thing is, to darn your darn socks. It makes them last a little longer, which makes the money last a little longer. Let's hear it for longer lasting money!