Once the harness is installed the power from the headlight switch only serves a trigger voltage to pickup the harness relays and keep them picked up as long as the headlight switch is on. Make sure you have the harness ground wire making a good connection to ground so the relay will work properly.
Here is how I would start troubleshooting this problem.
Turn on the headlight switch to pickup the harness relays - that should make the LEDs burn.
Then turn off the headlight switch.
If the LED headlights keep burning, unplug the plug that connects the harness to the original factory wiring.
The LEDs should turn off.
If they turn off there is a problem with the original wiring that is supplying power to the harness relays even when the headlight switch is turned off.
If the LEDs do not turn off there is a problem with the harness.