Hi there, astartz!
While we Sling folks don't support that configuration per se, there's absolutely no reason why it shouldn't work! We definitely haven't blocked those sorts of connections (you can do it with the iPad), so I don't see why whatever's on your Kindle shouldn't project through HDMI!
Good luck with that!
The Slingbox Answers Forum Moderation Team
P.S. Feel free to let us know if it works!