How to use YX8122 led driver?
A dim LED can be powered from a button battery cell for a short time.
I have some cheap solar garden lights that use a 1.2V AAA Ni-MH battery that is charged by a 30mA/2V solar panel all day. Its QX5252 IC is similar to yours but has an English datasheet and it lights a single LED or a colors changing IC brightly all night.
My AAA battery is MUCH larger than your button battery and it is charged all day every day.
Some LED novelties use three 1.5V button batteries in series and a resistor to limit the LED current. They light for a few hours.