Umm....from looking at your diagram...Are you actually trying to push the wall AC straight into the 7805?
The 7805 only regulates DC current, not AC. You need an 8-12volt (output) transformer. This is probably why nothing is working, and why the 7805 is heating up so much.
It takes alotta abuse to cook a 7805, so it may still be good...
Also, do you have the capacitors on each side of the 7805? They're kinda important too lol. As is having a resistor hooked in series with the LED to keep the LED from burning out. I'm guessing you'd probably need somewhere round a 160-180 ohm resistor in there to keep future LEDs from burning out.
