One of the three wires will have two quick disconnects (QDs), one of them goes to the microswitch and the other goes to the LED connection -- not sure if that wire is 5v or ground.

One of the two remaining wires go to the microswitch and the other goes to the LED.
LED QDs are usually 0.250" and microswitch QDs are usually 0.187".
If you've got the button press registering, it sounds like the LED is installed backward.
Scott