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You don't have to do anything. Everything is stored automatically 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. You will see your books under Cloud section on your new Kindle.
You can even download Kindle on your PC or phone meantime to access your books --> https://www.amazon.com/gp/digital/fiona/kcp-landing-page
Make sure you register the new Kindle using the same account used on your old Kindle. If you have any questions or would like any additional information, please let me know.