This info was not included in your original post. Had it been then the diagnosis changes: The micro USB-B port needs to be replaced. MicroUSB ports are designed for failure, thanks to their small size & usual lack of any additional means to secure them to the board (I notice your model does have additional clips that secure it). After a lot of use the insertion force required causes the solder joints to fail or the pins attached to the socket to break. [MiniUSB ports that are slightly larger and have a “T” shape are much more durable.]
MicroUSB-B ports are mounted (soldered) onto the board by precision machines, and not replaceable—however if you are adept at soldering you could get a center mount port & try it. Otherwise the guide for replacing the mother board is here: Kindle Paperwhite 1st Generation Repair
That is assuming you can find a MoBo [Motherboard]. These cheap electronics are meant to be disposable so, you could try to replace the port knowing you might make a mistake and end up having to buy a new one. Just make sure you get the right microUSB-B center mount socket (shown in the last picture of the guide).
So to recap in order of easiest/most expensive:
Replace the device
Replace the MoBo
Replace the microUSB-B socket with a possibility of having to replace the device
Again, good luck, and let the community know how it goes. Thanks.