this is such a long project for something that is sooo simple.
120V relay, on 120V line. end of story. spend the extra 30 cents.
switching the battery pack to turn on the LED lights is also a no-brainer.
no capacitors, no resistors, no microcontrollers. just one relay and some wire and Bob's your uncle.
well, SAVE 54 cents !!
if this project takes you more than 5 minutes, you are missing something very-very simple.