That series of switches includes ones without LEDs; they do not require power to work (other than the "latching toggle" one).
The LED is a totally separate indicator lamp in the momentary type.
The switching device is triggered by voltage from a piezo transducer & also possibly powered by stored energy - they probably draw a few microamps when open circuit & charge a small capacitor to hold the switch on for a short time when pressed.
Once you have found the LED wires, ignore them for now.
Just use the other two. Connect to power with a simple load in series - eg. a 24V led & resistor combination.
Try both polarities in case it only works one way.