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If you have no POWER LIGHT.....the unit power supply has failed.
There is not much a person can do about this problem, short of replacing the whole unit.
Inside every Electronic device that uses a computer chip (almost every electronic device in a home theater setup) there is a AC to DC power supply converter. The unit takes in AC power, and converts it to DC power to send to tall the internal PCB's and other components including processing chips.
These types of PSU's (Power Supply Unit's) are very susceptible to HEAT. If not properly kept cool, they will very easily burn out.
Once a PSU burns out, they are NOT repairable....they are simply REPLACED.
IN your case of a SLingBox....they dont make CONSUMER-ready replaceable PSU for a SlingBox...Unfortunately!
The SlingBox company position, is such that in the event of a Unit failure (of any nature) then the unit is to be Fully replaced....not repaired.
I recommend to you that you get a WARRANTY service done.
If not under warranty, you are stuck with getting a replacement.
Although if one were so inclined....one COULD in theory open the SlingBox....and remove the burnt out PSU, and replace with a new (pulled out from a different, working, SlingBox) PSU. Although you would have to be somewhat familiar and a Technically savy person.