The battery from OWC arrived in the mail.
These batteries are really thin. They make the MacBook as best they can, then measure the spare space inside the case and come up with a framework containing a bunch of really skinny batteries. However much they can fit in, that's the battery power available.
The box included the two tools I needed to get the case apart, and the battery out. It was pretty easy to open it up and remove the battery. I made one mistake, I forgot how the battery looked when it was connected to the motherboard. I couldn't get it back on when done. I had to bring the OWC video back up on another screen and see how the connector was attached. Once I saw that, I could clip it on properly. Then I screwed the battery down, and put the case on. The case felt as if it was bulging on the left side, so I took it off again, undid the battery and reseated it all again. I screwed the screws back in the order that the video suggested. When done, it still felt a bit bulgy on the left, but it all screwed done nicely, so maybe it just needs some time to adjust.
Flipped it over, turned it on, and the MacBook booted up and the battery had a 12% charge. So far so good. At least I didn't blow it up.
Then came the lengthy battery calibration. I had to charge the battery to 100% and then leave it charging another two hours to make sure it was at 100%. Then set all the Energy Saver settings to off, and let it run down to zero. That took a long time. Once it was down to near zero, the MacBook shut down by itself. I had to leave it draining for another five hours to make sure it was down to zero. Then charge it up to 100% again, put the Energy Saver settings back to normal, and then I was done.
It works. The MacBook is full of life again.