GFCI outlet with led light not on but have power
The light should only come on when the GFCI is tripped in most outlets.  There are GFCI outlet testers available; big box home stores carry them.  If the light doesn't come on when the outlet is tripped and the outlet tests correctly, it's probably the outlet.  Replacing the outlet would be the way forward to fix that.  The GFCI tester may show other problems which could cause unexpected behavior and should be fixed.