It's working again!
When I changed my mobile phone, probably through not following the correct procedure, I ended up with a band that was neither use nor ornament because it would not connect to my phone.
There was a fix on the internet, which involved discharging the battery, putting the module in the fridge for a few days, then trying to connect again. It didn't work.
So a few days ago I thought it was time either to sort it out once and for all or to send it off for whatever recycling is possible for these things. So once more I ran down the battery (it was already flat) and put it in the fridge for about a week.
Then I tried connecting.
"You need to factory reset your band" came the new message. I did that.
Then with just a tiny amount of jiggery-pokery - IT'S WORKING AGAIN!
I'm pretty sure that a few software updates from Xiaomi have made the difference.