Contacted Amazon by email at around 11 am today. Had a phone call from them around 3 pm - a very nice chap saying that although my Kindle was out of warranty, as it's a battery life problem they would send me a new one anyway. So that's in the post and I have 30 days to send the old one back. A confirmation email arrived a few minutes later.
I call that a result and all credit to Amazon. Their taxpaying record may be a bit dodgy but I can't fault their customer service.