The LED is working on the AC only during 1/2 of the AC cycle. If you look at it on a scope, you will see that it is blinking at 60Hz. You can somewhat see this when you have the two lead bi-color LEDs. On one polarity of DC they are green, the other red, on AC, each of the LEDs conducts 1/2 the time making it kind of yellow.
The reason the resistor gets hot is that it is underrated for your application. You are dissipating about 1.5W in that resistor (P=V*V/R). The voltage across your resistor is about 118Vrms or so. I dont know what the codes are in your area, but I would suggest using a 120V indicator. In any case, a larger resistor (2W would work) is needed if you go this route. At 9.1K you are supplying about 13mA to the LED for 1/2 of each AC cycle.