The "ipod" input is just usb. It can play music from any old usb drive. I imagine if you can plug the kindle into a pc and play the music from the pc you'd be able to play music from your kindle thru the reciever. Of course if the kindle has a port for headphones, it will most defintely work. In either case you likely won't be able to control the kindle thru the reciever remote. Hope this helps.
If you are talking about connecting the Kindle (assuming fire HD) to the reciever, you can use an HDMI cable and connect the Kindle to the reciever. With that, you can listen to music, Netflix, photos, etc... just like connecting your smart phone to the reciever.