The circuit is fine. Heed what PCB.Wiz said.
A neon bulb is the right tool for the job and saves on some calcs, some power and some components, but the LED is easier to tune and is more familiar to many these days.
The challenge with mariush's circuit is that that cap gets subjected to mains voltage. There will be an enormous in-rush, and if it fails it shorts your supply. Nothing that can't be managed - every rectifier is a variant of this - but it wouldn't be the first option I'd go playing with if you only want mW of power.
For the LED circuit, consider the role of the R and C. Together they limit current. At PCB.Wiz says, 5mA will probably be plenty. So you need R = 55/0.005 = 11kΩ. At 5mA, power in 11kΩ is 0.005*0.005*11000 = 275mW. So you could use a 1W resistor and call it a day.
But it's a bit wasteful. So suppose you added C = 0.33µF. Zc = 1 / (2*pi*50*0.33µ) = 9600Ω. Drop your R to 4.7k and you get total Z = sqrt(9600*9600+4700*4700) = 10.7kΩ. Similar sort of current but now the power in the resistor is just 0.005*0.005*4700 = 120mW, and a half watt resistor will be fine.
Also note you can replace the 1N4007 with another LED if you want to simplify you BOM. Now your 5mA is making light 100% of the time instead of 50% and just making stuff hot the other 50%.