First, & very importantly, LEDs are INHERENTLY DC devices. You have a
string that uses 120v AC because there is a rectifier built into the
string, or each bulb has 2 LED's, in opposite orientation.

So, you need to determine which it is. Here's how: get a DC supply of
more than 3.1v. You can use 3 AA or AAA batteries, end to end, or a 6v
DC wall wart. Connect a 100 ohm resistor in series with the supply.
This limits the current to 30 ma, so as to not burn out the LED.

Then connect the supply to the LED, both ways (both polarities). If the
LED lights with both connections, there are 2 LEDs in your bulbs and
you'll need an AC supply. If it only lights on one polarity, there is
just 1 LED and you'll need a DC supply.

Report back with your findings and we'll go from there.

Bob