It actually requires two diodes. In the stock wiring system, the polarity for the current switches depending on whether you're signalling left or right. The incandescent bulb doesn't care about this, but an LED bulb definitely will.
The fix is to disconnect the orange and the blue wire from the indicator bulb and solder (or otherwise attach) a diode to the end of each. The ends of each diode are now the positive side for the LED. The negative side should be wired directly to ground. The diodes are necessary in this circuit to prevent the current from bypassing the LED back through the opposite spliced wire.