Hi and Welcome. My Name is ***** ***** I look forward to assisting you today with your question.
All your books are saved on the Amazon account (i.e in cloud). All you have to do is register the new Kindle with the same Amazon account you used on your old Kindle and all the books can be redownloaded. I assume you already did this part. So you will see the purchased books under Cloud section, in Books. Open Books, tap Cloud and you will see the books there. Download them. You will not be charged again.
Another way is, on your computer, goto Amazon's Manage Content and Devices site i.e go to this website --> http://amazon.com/myk
Log in to your account. You can find all your books there. Just send or download them to your new Kindle (deliver the book to your Kindle using the 'Actions...' menu to the right of the title).
If you have any difficulty, please let me know and I will walk you through it.
Remember, this is not my final answer. If my solutions didn't work please don't rate me bad or poor. Instead please reply back to me and I will continue to help you until we fix the problem or you are satisfied.
Are you looking on your Kindle itself (under Cloud) or the Amazon Manage Content and Devices website ?
You have to sign in to the original account you used on your previous Kindle. Go to this website and sign in with your correct Amazon account --> http://amazon.com/myk
Do you see your purchased books there ?