I fixed the sock!!! And I didn't even have to frog it!!! Frogging most likely would have been easier, but if I had ripped it out I definitely would not have had the motivation to make it back up again, so I went with the picking up the stitches approach. It was actually not nearly as bad as I thought it would be!
In case you've never reknit just a section of a knit piece before, here is a step-by-step explanation of how to do it.
Step 1) Shown yesterday: Insert a double pointed needle into the last "good" row where you have identifiable stitches. Just pick up the stitches you are going to be reworking.
Step 2) Shown yesterday: Drop the stitches from the top working needle. Make sure that all these stitches are going to be caught on the bottom holder needle. This is terrifying!
Step 3) Look closely at the ladders you now have in your knitting. Find the one that makes up the row above the stitches on the holder needle. This will be the working yarn for your next row.
Step 4) Now you just pretend now that you are knitting like normal, with this ladder yarn as the working yarn. I have no idea how to do this step using the English method, but here's how to do it Continental. I suspect it is much easier this way.
Step 5) As you get to the end of the section you are working on, you will be working with a very teeny piece of yarn (hopefully your yarn piece will be the right length for the part you are knitting!). This can be tricky, but you'll figure out how to maneuver the little bit of yarn to be a stitch. I just pinch it and force it to go where it's told.
Step 6) Once you finish the first row, look to see if the ladder yarn you just knit with is evenly distributed across your new row of stitches. Is there some extra yarn that is making a giant space after this new row of knitting? I tried to even out the tension by slipping the stitches back and forth between two needles a couple times pulling and tugging the yarn a bit until all the stitches seemed to be about the same size. In these socks, the new row is still a bit loose (there is enough yarn for a whole extra stitch in each row since I ripped it out to remove an extra stitch) but it shouldn't be a big deal once I wash and block them. Because, again, there will be no frogging on these socks, darn it.
Step 7) Repeat steps 3-6 until you are way back to where you started, only with a perfectly fixed up piece of knitting!
Something to watch out for: be sure that the stitches you pick up in the first step are in the correct row (ie, don't pick up stitches and knit row two when the rest of the row is row three!). I made this mistake. I don't care. No one else will care, either. You can't even tell.