Servo and led control with button not working.

int ledPin = 13; // the number of the LED pin

pinMode(ledPin, OUTPUT); // initialize the LED pin as an output
ledPin=LOW; // we starts with faceplate open and eyes off

Changing the value of the pin number has nothing to do with changing the state of the pin at the original number.

That's what the digitalWrite() function was invented for.

You've said nothing about how the switch is actually wired to the Arduino. That is critical to being able to read it properly.