I have had a dreadful time trying to get our Kindles to charge, update, or in general, work, and it all comes down to the cable.
Whilst it has a standard micro-usb connector, the cable has some proprietary jiggery pokery in them to keep them locked down.
Of course, even brand new cables were doing nothing for our machines, and it was only when we got a whole new unit, with official cable, that we got the thing working.
To prevent this in the future, we have produced a wee tag to keep the cable recognisable in the ridiculous collection of cables and cords we have accrued.

This file is designed to be printed flat without supports. though you will need to shave out the channel on the open side in order to pull the cable in.If you have opened it up too much, you can close it up again with hot glue, or wrap the cable in electrical tape to thicken it up. Hope this helps someone!