No, you definitely do not want to run high voltage (120) with low voltage wiring. I mean it can be done, but it is not safe and can shock you or damage your equipment, so I say don't do it.
Looks like that relay will work.
And an led should be fine but you may need to run three wires, one +5VDC (can probably get it off the usb port).
A ground for the LED (also off usb, possibly. usb is good for about 500 ma, so figure what kind of load will be on it.)
And the wire back to relay to turn it on.
Wire +5 to your switch.
Other side of switch goes to the relay, also to the led.
Other led wire goes to ground. (Also suitable resistor in line with led.)
I can give you a simple diagram if you need it.