First replaced the latch, spring, and lever, but no improvement. Then I saw the lever arm on the other half of the door, which by position had to be the way that the timer interacts with the soap dispenser. After turning off the power to the dishwasher, I removed the switch cover (not obvious how -- a video would help: on the left, I slid the narrow rectangular plastic wire cover a little to one side and gently opened it on its little hinge, which is near the left side of the door, and then I was able to slide up and lift the plastic cover for the switch. I removed two screws from the top of the switch, and it lifted out. There I saw that the plastic arm had a spring dangling loose from its end that had been hidden under the switch cover. I also saw that there was a plastic post on the door that was supposed to anchor the loose end of the switch. Inspired by other advice on this site, I drilled a hole into the middle of the base for the broken post, using the narrowest drill bit that I have. I had to scrape away plastic curls melted from the drill, and then I gently but firmly inserted a 5/8-inch wedge nail that just fit in the hole, using my thumb. I didn't need the super glue that I'd thought, because the nail fit tightly. This is like a finishing nail, only it's got a bit of a flat head so that the spring can't slide off the end. I hooked the loose end of the spring into the nail, put back the switch and its cover, and closed up the door the same way that the video for replacing the soap dispenser latch showed. Working great! Took me a long time because of investigating, but if I'd worked the problem systematically and known how to disassemble the switch, probably 20 minutes would have been enough.
Oh, and I also replaced the door spring. A video would be helpful for this, too. Used only a screwdriver, and it took about 30 minutes because I had to figure out how it was put together. By the way, this washer has two cables and two springs -- I recommend buying all four parts if you're going to get into the problem anyway.
People on phone at AppliancePartsPros were patient and well-informed. I'll go back for future repairs.