You say "one of the lights on the front of the Slingbox cycle." Do they come on at all? ... or is it just the reset button, when pushed, the LEDs don't cycle?
Clear the slingbox from the router and see if it will connect to the router.
If the reset button is non-responsive, it sounds like the internal electronics.
Is it brand new?
How old is it?
Hooking up after a while off?
While powered up ..hold the reset button for more than 30 seconds...
Disconnect power for a minute or two
You have reset it back to factory and need to run set-up again and connect it to your network using SlingPlayer