The Apple TV's Mini USB port is solely used for service & diagnostics. That being said, you must be trying to restore your device from iTunes, which is the only USB function I'm aware of. I don't know why you'd connect a Kindle? Perhaps this is what's confusing me. Could you be more specific? Sorry, I hope I can understand & be of assistance. How is the device behaving? what's the issue?
Here is a guide to restoring your Apple TV. If that's your goal, I'm sure you found it already: http://support.apple.com/kb/ht4367
If that does not work, then I'd assume it's a hardware issue. The USB cable, or the USB port on either device may be faulty. at which point, I'd say vazandrew gives good advice.
Wish you luck! 😀