When booting your PC choses it's connection type, VGA/DVI/HDMI. It might be trying to boot with VGA if you boot without plugging in and turning on your TV first. Try this:
Ensure both PC and TV are OFF!!!
Turn on your TV only
Select HDMI 2 in the source
Make sure HDMI cable is plugged in to PC and TV
NOW turn PC on.
If you turn your PC on then your TV and are using HDMI as your source, the PC might find no device on VGA, try HDMI, nothing there default back to VGA then you turn on your TV. It doesn't switch automatically, its outputting on VGA despite nothing being connected there (as this is the most common connection option for PC's) and leaving the HDMI blank.
Also your Kindle can be seen as a screen option and the resolution could be changed to fit the Kindle and it might be using one that does not display on your screen. Try using splashtop "with the devices native resolution (as in your PCs, not the Kindles) instead of a predefined one (like 1280x720, or 640x480) or instead of "fit to device" (as in the Kindle).
Also try without any other displays attached so make sure VGA is all disconnected, any DVI ports empty and even your Kindle disconnected from Splashtop if not resolved above. Then after that use Splashtop.
Also it most certainly could be resolution. HDMI uses 800x600 only on HD TV's. I found this out once before. So if all else fails, connect your Kindle in this mode, see if it then shows up on your screen. Let me know if and how you fixed it as I am interested.