I will use the conventional solution of the rectifier followed by a capacitor, with a circuit to limit the load current of the latter, as explained elsewhere. Since the LED consumption is very low, the capacitor’s value may be relatively small. After tests, I found that 100 µF 35 V was ample enough, but I kept the 470 µF capacitor initially planned.
Here is the diagram of this circuit, which will simply be inserted between the wheels on one side and the original circuit on the other. I made this circuit on bakelite stripboard that I had at hand.
Here is a 3D drawing, prettier and more readable than the real prototype circuit! Pay close attention to the capacitor polarity! Provide fairly long connecting wires (10 cm) for easy connection. It will always be possible to conceal the surplus into the hopper after reassembly.
In the plan view, the capacitor is rendered translucent and the other components are represented by a white outline. The tracks are seen by transparency, they are actually on the opposite side of the board. They must be cut as indicated (under diode, resistor and rectifier bridge). Connections to make (solder bridges or bent component wires) are indicated by red lines.
The connections are, from top to bottom, and in the same order as on the circuit above: