If the charger's light doesn't come on at all — not even a flicker — even when you've tried a different, working outlet, that's a pretty strong sign the charger itself has died. The light comes from the brick, not the Surface, so it should light up on its own even before it's connected to anything.
Since your Surface is less than a year old, it should still be under Microsoft's standard one-year warranty, and that usually covers accessories like the charger too.