I fixed the toilet! We now have two flushing toilets in the house! Huzzah! Huzzah!
I was going to work on the situation on Saturday after buying the toilet guts, but discovered that the shut-off valve doesn’t completely turn off the water. I hemmed and hawed about it and then called Brad. “Just work fast,” was his gifted advice, and he was right. I just put a bucket underneath and prayed that it wouldn’t make a huge mess.
It all worked out. At least, it all seems to have worked out for now. There’s a towel on the floor to let me know if there’s a leak I missed. If I missed something, I’m just burning this house down.