Three seconds of research on the interweb gave me this idea: proof some new yeast that isn't dead, and mix it into the existing bread dough. Then start the rising process again. Specifically: put a packet of yeast (2 1/4 teaspoons) in a bowl with 1/4 cup of warm water and 1/2 teaspoon sugar, which helps wake up the yeast. Walk away for 10 minutes. When you come back, it should be super bubbly and smell kind of sour. Knead the liquid into your existing bread dough. (This is messy, since you're basically pouring water onto dough; it wants to run off. I just did it a little at a time.) I also sprinkled on a little more flour so it wasn't completely sticky, but I didn't worry about re-kneading it to a satiny finish because I didn't want to overwork the dough. Re-spray the bowl with cooking spray, turn the dough to coat, cover with a towel, and let it rise for an hour and a half.
When I came back, 90 minutes later, it had started working! And after punching it down and giving it another 30 minutes, the dough was totally back on track! I just braided it and left it for the final rise. Other than the fact that it should've been done hours ago, it looks perfect!
For what it's worth, the yeast I used to fix it was the same yeast I used in the first place, so clearly it wasn't dead. I have no idea what initially went wrong, but I'm so excited I could fix it! I'd hate to waste all those good ingredients, and I was REALLY craving fresh challah! (I'm making roast chicken and potatoes and broccoli, too... mmm...)