This problem sounds like a hardware issue. Before my final answer, let's check every possibility to be sure it isn't something wrong with the software.
Please follow the steps below to reset your device:
1. Press and hold (or slide and hold) the power button for 30 seconds.
2. After 30 seconds, release the power button.
3. Wait 2 minute, then turn on your Kindle.
4. Try to charge your Kindle.
Please try this procedure at least 3 times (if you have leather cover or any other cover remove it).
If the previous procedure doesn't work, perform the procedure below:
1. Plug the Kindle into the charger for 5 minutes.
2. After 5 minutes (while still plugged in) press and hold power (slide and hold) button for 30 seconds.
3. Allow up to 3 minutes for the Kindle to reboot.