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Those models are known to have a bad charge port.
If that is the case then the best option will be to get another Kindle.
unfortunately this does sound like it might be a hardware issue. We will need to rule out the possibility of a firmware issue by resetting the software, using a force reset method while USB charging. If this fails then there is an hardware issue on the device that software troubleshooting can't resolve. Try these steps please>Connect the usb charger to the kindle and a wall outlet charger. Then Slide or press and hold the power switch for 40 second, then release. Wait 15 minutes then power on and check. If that fails and you are sure that the charger is working properly then that means the kindle has a hardware failure.Usually the cost to repair is too much and the best option is to purchase a new or refurbished kindle which is a good value,
You can read the books from another device or computer while you are getting the kindle replaced with the Kindle app which is available for most major smartphones, tablets and computers. That means with your free Kindle reading apps, you can buy a Kindle book once, and read it on any device with the Kindle app installed.