LED Help

I think the easiest way would be to eliminate that board and get a regulated DC supply to replace it. You seem to have three wires that go to the LEDs. If you operate the button to get to the steady state condition and use a meter to see which wires have power and what is the voltage that would be the information to choose and wire the new board.

yeah, we need to know what's on the other end of those wires too. Voltage/amps in/out etc...."whatever you got".  The 220v also throws up a flag of caution. If this is indeed functional to 220v, I'm not sure you'll like each other. 

Offhand I'm guessing there are actually two sections of leds. The L+ goes to both banks of led; the Q1 &Q2 are the negative legs for each bank. (most  often, electronics tend to use the ground path for switching, and a constant hot) Q is a transistor.

Test an led with a dc source like a battery(s) (AAA,AA,C,D are all 1.5v ea,)start low at 1.5v, then 2= 3v, then 3= 4.5v , etc, etc. If it lights dim try stepping up voltage a volt or so at a time, but mind that even a slight over voltage will burn out an led in an instant. (They usually light pretty well while fed way under the max voltage. 

Solder jumper wires over the push button?

Old Goat

I'd eliminate the board and just wire the LED strings with the appropriate power.