It sounds like whatever the switch is that is connecting the router to the living room isn't passing DHCP normally, or it is reserving IP addresses and then possibly reassigning them to multiple devices. If you have problems with the TV using the ethernet in the living room, you'll most likely have problems with the slingbox - or any other device for that matter. What is the exact topography from the router to the end of the ethernet cable in the living room? Can any of it be reset, rebooted?