Hello. My name is ***** ***** I can help you with your problem. Kindle for PC is an application that emulates a Kindle. It is a destination for your books, but it has nothing to do with transferring content from your Amazon account to an actual Kindle.
Your Kindle Fire should show up on your Amazon account when you access it at www.amazon.com from the internet browser on your computer. You can connect your Kindle Fire to your computer with a usb cable and transfer the books, one at a time, by checking the box to the left of the book and selecting the action box. Select the Download and Transfer Via USB Cable action.