Presumably, there is a low voltage power supply inside, run the LED off a low DC voltage.
Bob
For a LED it would be easier to run it off of an internal low-voltage line.
Instead of a LED, you could use a neon lamp with a suitable capacitor. The little rice-grain neon lamps are usually rated for 220 VAC, but they still do light up when connected to 115 VAC.
The neon lamps glow orange, so you don't really get to pick the color. They do have a very retro look about them. You may find yourself shopping for 3S4 and magic eye tubes eventually. There is something fascinating about watching electrons do circus tricks out in the open, where you can see them.... But anyway.
Also it is helpful to explain, if you did not know:
An LED is a light emitting
diode, but
it isn't made to work real well as a diode, and so you shouldn't use it that way. It's made to light up well.
Most visible LEDs can't withstand more than about 20 milliamps forward current, and can't withstand more than 10 or 20 volts reverse voltage. That's pretty lousy numbers for a diode.
Meanwhile, a very cheap and common 1N4007 diode is smaller, can withstand 1 amp forward current, and is rated for 1000 volts reverse voltage. And it's nothing special.
So if you must run a LED + resistor off an AC source, then you also want a 1N4007 diode in series too. Because that LED won't live long on AC power--even low voltage power.