Mostly for educational purposes, you
can make such capacitor circuit to work, but again practicality is questionable. Rough process:
1. It is best to take full bridge rectifier. Look in datasheet, what maximum pulse current it tolerates.
2. Choose current limiting resistor R1, so the maximum possible current does not exceed diode bridge maximum allowed. Lets say around 10A.
3. Choose capacitor C1 to set LED current.
4. Choose capacitor C2 to be at least 200x bigger then C1, so that C1/C2 divider does not exceed LED forward voltage. Also it needs to be low enough ESR such that at peak pulse current, the drop voltage does no exceed LED forward voltage.
It should work reliably, but if your LED becomes open, the capacitor C2 will go overvoltage. So LED should always be connected.
Edit: Added R2 for high voltage capacitor discharge, as suggested by SiliconWizard
