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There are mainly two reasons for this problem. Either this is an firmware issue or the battery has failed. In some cases issues with mainboard also cause this kind of problem.
To make sure this is not a firmware issue, first we can can soft-reset the Kindle and see if it works or not. Press and hold down (or slide) the Power button for 20 seconds and this will reset the Kindle.
Please let me know if that didn't work, so we can continue troubleshooting.
If it still not turning on, connect the Kindle to your PC with the USB cable, wait 2 minutes and then hold down the Power button for 20-40 seconds. Let me know how it goes.