A quick update ....
Although I'm not too bothered about having to replace the Kindle, I thought I would talk to Kindle Support, as per Andy's suggestion.
When I pointed out that the Kindle was out of warranty, the helpful support guy said that they would be willing to replace it anyway, but not for free. There would be a charge, which would be less than the cost of buying a new one.
He said that someone would contact me within 24 hours to let me know how much the charge would be, at which point I could decide whether to go with it. That was yesterday.
Today, I received an email saying that my new Kindle was on its way, and would I return the old one within 30 days. Nothing about how much it would cost.
I'm not sure whether to believe it. I'll let you know what happens.