The stand light looks like this:
The only information on the battery is 3.7 volts and it looks like this:
My question is, how can I figure out what voltage and current rating the power adapter needs to be in order to power the light and charge the battery efficiently? If I put 4.0V on the input jack without the battery installed, the lamps seem to light up fine. If you look closely inside the battery wrapper, you can see an attached pc board. The wiring from the DC jack goes to a series diode, then a transistor, then to the red wire extending out of the neck of the lamp where the LED's are attached. There is also a blob of round black stuff on the main pc board that looks like it could be covering a uP.
Thanks for your replies.