Fixed the problem
Apparently these micro USB sockets are a weak link in many devices, and the one in our Kindle Fire gave out in about 2 years - in common with many others. Replacement of the socket is a specialist job - don't even think of tackling it with a soldering iron - and you're most likely to find this service as a shop that deals in mobile devices rather than computers. Think around the £40 mark. However, it's all fixed now and when the lead is pushed in the socket connection feels nice and firm. How long it will last, who knows? But better than buying a new Kindle.
Verified purchase: YesCondition: New