What am I doing wrong and what should I be doing to fix this problem? My friend wants to use a 12V relay to solve the problem but I think that is excessive and wasteful. Thoughts?
I think your friend is wise and resourceful. 
The poor relay gets so little respect in the arduino world.
How about another solution, use a DPST (or of course a DPDT) switch for your main power switch and use one side for applying or removing AC voltage to your power supply and use the other side of the contacts to route the +12vdc on and off to your led(s).
The choice of relay or switch being the best choice depends on the physical location of your AC power source, power supply and existing switch locations.
Lefty