I have tried everything in the troubleshooting guide...
Cool. I'll skip that stuff then.
Your Airport Extreme should have nothing to do with the problem if other devices are successfully using it. If you haven't already, try connecting a different device to the ethernet port your Slingbox is currently using. This will verify proper router operation and physical ethernet port function.
If you've power cycled the Airport Extreme and performed a hard reset on the Slingbox without success (sounds like you have), then it's quite possible you're looking at a Slingbox hardware failure.
The light will not flash during startup, no light at all times.
From your description, this is my primary clue for hardware failure, along with your troubleshooting up to this point. You should get some activity on power up, even if it's brief. You should definitely get some activity after a hard reset.
A number of people have reported success by replacing their power supply. Your description doesn't quite fit the profile though. PS failure is often indicated by weird noises or excessive heat on the PS, and dim/inconsistent lights on the Slingbox. You mentioned none of those. There are a number of threads on the board about this, search for "power supply" or "power adapter" if you would like to check them out.
Here's a thread that has the PS specs for your Sling Pro:
Sling Tech Support can probably assist you further, but you'll have to pay for the repairs since you're out of warranty.
Wish I had a more optimistic assessment. Good luck !