I switched the computer on this morning and left it while I made coffee.
When I got back, there was a message to say that Windows had found new hardware.
I connected the Kindle and... nothing! However, I left it connected to see if it charged. After about 5 minutes, Windows found the Kindle as if nothing had been wrong.
So, thanks all: I'll keep this topic just in case.
This thread is now locked and can not be replied to.