This is an old question and the answer is just for completion sake.
This “. It appears the input connection in the Kindle is worn or otherwise won't make good connection with the charger connector” would indicate that the connector needs to be replaced. It is soldered to the motherboard. You will have to remove the mother board. Use something like this video to remove it. After that check the solder connection on the micro USB port. It is very likely that they have either come totally of the board, or that they have micro cracks. Next you will have to desolder your old connector and replace with a new connector which requires soldering again. . If you do not have the tools or the skills (yet) I suggest you find somebody online that has that experience. Soldering those connectors is a bit of a PITA. Ports are a bit hard to find so scour ebay.com, digikey.com or mouser.com and see if you can match it up.