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Have you tried different charger/cable ? There is a trick to reset all the hardware components of the Kindle. But if the issue is with the charging port or battery, then this trick will not work.
To reset, Press and hold down (or slide) the Power button for 20 seconds and this will reset the Kindle. Then check if you are able to charge it.
Please let me know if that didn't work.
If the reset doesn't work and if you have already tried different cable, then it means either the battery or the port has failed. You will have to send it to Amazon to get it replaced.
Amazon don't repair the Kindle. They usually replace the Kindle with a new one (if it is under warranty) or will replace it with a refurbished one for a reduced cost if it is out of warranty. To setup replacement, you have to contact Amazon. Here is the number --> +1(###) ###-####or this number --> 1800 3000 9009
You can also contact them from here --> https://www.amazon.com/gp/help/contact-us/general-questions.html?skip=true#b
You have to send it to Amazon. Call Amazon and they will arrange the shipping and send you necessary details.
Or you can also buy a new Kindle. Everything is stored on your Amazon account in cloud. When you get a new Kindle, all you have to do is register it with the same Amazon account you used on your old Kindle and everything you purchased can be redownloaded without paying again.
If you have any questions or would like any additional information, please don't hesitate to ask.
This is indeed an hardware issue. Holding down the Power button for 20 seconds is the only trick to reset its components. You will have to send it either to Amazon or you will have to buy a new one. But you will not lose anything as everything is saved on Amazon account. When you get a new Kindle, you can redownload all your purchased books without paying again.