This circuit should do the job and requires only an additional 400V 100nF cap. Any value around 100nF would do.
ERRATA
(1) 4K7, 2W resistor should read 4K7, 4W
(2) VF=3V should read VF=3.45V in two instances
(3) C6 schematic symbol should be for an electrolytic capacitor with the positive terminal uppermost
(4) LED2 & LED3 should read LED2-1 & LED2-51 respectively
NOTES
(1) The input capacitor and two diodes in the bridge rectifier are a voltage doubler, generating 170V across capacitor C7 on the negative excursion of the mains sine wave .
(2) Some of the energy stored in C7 is then transferred to the LED string on the positive excursion of the mains input sine wave which adds the positive swing of the mains sinewave to the 170V stored by C7. Thus the LED string is presnted with 170V + 170V = 340V (not quite true, but good enough for illulstration)
(3) The current through the LED string is pulsed, but the average current (RMS) through the LED string is set to 20mA by the series resistor, R9.
(4) D6 & D9 have no function and are only shown because they are part of the bridge rectifier.
(5) First build the circuit with just the input capacitor C7 and bridge. Ensure there is 170V across C7 (there will be no ripple voltage)