advice needed for mini solar powered electro luminscent light

Hi, I'm trying to put together a solar powered electro-luminscent light that uses a rechargeable lithium ion battery. As a way to experiment, I've taken apart an electro luminscent night light that plugs into a wall socket, and tried to hook up the wafer thin light to a 3.6 volt Lithium-ion button battery, however it does not light up. My questions are:

1. The specs for the salvaged EL night light is 120 volts, and .03 watts. Is the fact that the battery I tried to use is only 3.6 volts the reason why it won't light up? Why does the indigo EL light in a Timex Ironman watch light up ok with a 3.6 volt battery in the watch, yet the light I salvaged from the night light does not?

2. What would I need to do to power the salvaged EL light to illuminate with a Lith-ion button cell?

3. Would it be possible to use a 1 inch by half inch size solar cell to power a rechargeable lithium ion button cell that would power an EL light?

4. Are there any good book or web-based sources to learn about this sort of thing?

Thank you very much for any advice anyone can offer.