Ok, so I've got a sheet (8 years old or so) that has a big shredded spot. . . . So I mended it.
Here's the hole:
I read up on different methods of fixing sheets, and here's what I did:
I ended up cutting 12" square patches (one for each side of the fabric, because the area needing repair is totally shredded) out of an old cotton sheet that has even more holes in it, that I've saved for just this sort of thing. I then cut an 11.5" square piece of fusible web (think of it as big sheets of double-stick tape for fabric, except that it's permanent, and bonds with the fabric). I bonded one side to one of my patches, then folded over 1/4" of fabric and bonded that to the back side of the fusible web. I now had an iron-on-patch.
Hint: cut diagonal triangles off each corner so that when you fold it over, it forms a mitered corner. This reduces bulk at the corners.
I then cut the corners off the second patch and folded the edges of that in by 1/4" and then pressed the heck out of it. I then positioned the iron-on patch with the fusible side up on my ironing board. I then placed hole, top side down, centered on top of the patch. I then placed the second patch over the hole, trying to get the edges as even as possible with the other patch. I then used a wash-out fabric glue stick to hold the edges down, so it wouldn't shift while I was stitching it.
Then I ironed the heck out of it, fusing the web to the fabrics.
Next step - take it to the sewing machine. I stitched from the top side, as close to the edge of the patch as possible (trying to minimize the amount of "edge" that can catch on toenails or the like). I flipped it over, and found that I'd missed the edge of the lower patch alone one side, so I stitched a second row of stitching a couple of millimeters inside the first row of stitching.
Voila: I end up with this. One thing I would do differently next time is round the corners - less likely to catch the corners on stuff, and I think it would be smoother.
The sheet has been in nearly constant use for 8 years. It's a really high-quality cotton, and king-sized sheets are expensive. But... the fabric is pretty threadbare in places. I think I've probably only gained a few months of use, or perhaps a year or so. But, better than buying a new one, at least for now.
Edit: I slept on the patch last night, and while it's a little stiff, it matters not at all down near my feet and lower legs as that is. If I try to find it with my feet, I can, but otherwise it's completely fine.