The circuit should work. You should really have a resistor of 100 ohms or so in series with the 0.68 uF capacitor, to reduce the inrush surge as you turn on.
Transformerless circuits like that can be dangerous to fault find.
I can't really see why you don't have a smaller C1 and put all the LEDs in series. You then don't need R2 or R3. C1, combined with the mains frequency and voltage are what control the current. R2 and R3 are only needed to distribute that current evenly in the two strings of LEDs. You've got lots of voltage so 20 LEDs in series is still far less than the supply, so there is no reason not to do that.
Edit:- R2/R3 will keep the current more constant. However, with C2 there, the current won't drop to zero for a long time so the LEDs won't appear to flash.