I don't think it's routing. Can you SSH to the Kindle? Also, the behaviour when pinging the host PC would be different. Without a route, it would fail at once. If you just don't get any replies, there's probably a firewall on your desktop PC blocking the connection.
The viewer will just try to reconnect over and over, probably you have to kill it (or use the corresponding key sequence for launchpad).
BTW, the credits for the great description go to tinyapps.org!