The charging plug in connection is loose or broken

brian , this is really not a repair for the inexperienced because the charge port for your device is soldered to the mother board and the heat generated could cause further damage if not very careful. I will post a link to a tear down guide to get you inside the Kindle to check the charge port and will also post a link showing what is involved in repairing/soldering a mini USB attached to the board. The last link is to new charge port, which you can also do a search for better pricing perhaps if needed. If not wanting to do the repair yourself have a good computer/cell phone repair shop have a look and estimate repair. Good luck.

Kindle Paperwhite 1st Generation Motherboard Replacement

https://www.youtube.com/watch?v=ZWFMCIdZ...

https://www.amazon.com/StarSide-Charging...