I darned the socks! Huzzah!
Thanks to everyone for their excellent tips, especially Jan for her very detailed instructions, and to StitchBliss for the link to this video tutorial (which has a most amusing serious voiceover but which has an excellent level of detail!). Oh, and to Gae for suggesting to put a rubber band around the base of the sock-covered mushroom - that does make it a lot easier, by holding the sock in place on the mushroom (because it's a slippery little sucker!)
When I stretched the sock over the mushroom, I noticed a lot of weak spots around the hole, so I didn't end up sewing around the hole, but just started weaving through the sock, starting in a strong point and working from right to left to cover the weak spots and the hole, and ending on the stronger point again.
Then I turned and wove the other way, including the actual weaving portion over the hole.
All covered up!
When I turned it right side out, I found I had some strands that were not quite woven in as well as I would have liked, but all in all I'm pretty happy with this darning business.
I still hate sewing, but ten minutes of it to save a week of knitting - it's totally worth it!