AC current is not as simple as it sounds, you'll get flicker by just direct driving the led with a resistor inline. Best way to accomplish smooth lighting is putting a power diode aka rectifier inline and smoothing the resultant half wave output with a large power capacitor, I'm guessing something like a 1000uf 180v capacitor. You could reduce the capacitor requirement to about half with a full wave bridge rectifier using 4 diodes.
Say you want to run 35 mA to that single led. R=V/I so 110 volts minus the vf of 3.7v is 106.3, divided by the drive current of .035 A, this means you need about 3037 ohms of resistance so a 3000 ohm is good enough. but the kicker is that you'll need at least one rated for 4 watts. You're wasting lots of power in that resistor and more in the rectifier.
Trust me it's safer, cheaper, and more efficient to get a small transformer to drop the voltage to around 5 volts, usually cheapy usb power supplies do this, then you only need a small value resistor, maybe only 50 ohm's or so, and only a cheapy 1/4 watt resistor.
These are small switching power supplies that allow powering a small load from an AC outlet (i.e. mains power) without the use of a large or expensive transformer. It also has the benefit of being pretty efficient too.
On the other hand, I've used an old "wall wart" transformer to run a 3 watt Luxeon. It's not an elegant solution, but it is quick and cheap.