You would not get a forward voltage drop on the LED itself because the forward voltage drop is around 3V, and your meter likely only shows up to 1.999V in diode test, though it might put out up to 4V open circuit in that mode. However those LED strings have a typical forward voltage drop of around 100V, if they are all LED's in series, or a forward voltage of 3V if they are all in parallel. You can just plug it straight into 120V and see if it lights, and if so just use it that way, otherwise you will have to fit into the inner space a voltage doubler to run it off 120VAC.
As you already have one capacitor and a bridge rectifier all you need is another 15uF 400V capacitor, and unsolder the negative lead of the existing capacitor, add the new one positive to it, connect the new one negative to the hole you unsoldered from, and then remove the power lead not soldered to the resistor, and solder it to the junction of the 2 capacitors, and you have the voltage doubler that will light it on 120VAC.