An Arduino pin set as an INPUT is a high impedance input. So you do not need to "switch" it on. What is the high voltage of the switch that turns on the LED? If it is 5V you can just connect the Arduino INPUT pin to the same point as the high-side of your switch.
And do both circuits need to be powered from the same power source for this to work?
Voltages are based on references, so all components which are connected to each other much share the same reference. In this case, it means the GROUND of each "circuit" should be connected together.
GROUND is another way of saying "reference plane."