If you want to hack the existing power supply board, you could... but there's cheap alternatives out there.
See the image I attached, with annotations. And see bottom of the text for the alternative with links and all that.
The top section has the CCFL backlight controller section - The controller chip is that BIT3105 which has an operating voltage between 4.5v and 13.2v. On most monitor power supplies I saw (old Samsung and LG monitors with CCFL backlights) the voltage for the CCFL section was around 18-20v but it's possible this particular board uses only 12-13v.
Anyway, the capacitor circled with yellow/green should be on the input voltage path so in theory, you could solder there the 12v and ground wires from a power supply and then CUT the traces coming to the capacitor to isolate the section from the rest of the power supply.
I'm not 100% sure as I don't see the back of the power supply board.
As for the rest, the power comes from the left, goes through those fuses (round brown/red things), then through the input filtering (cream/yellow rectangle and stuff below my yellow text) , then goes into the bridge rectifier (the black rectangle to the left of that big transformer with green text) and then you have a bulk capacitor which smooths out the rectified AC voltage.
There's a controller on the back side which controls the mosfet or transistor to the right of the big capacitor in order to send pulses of electricity through the transformer, and on the other side comes out 2 or 3 voltages ... either around 13v and 5v or 5v and 3.3v
The two chips to the right of the square transformer on the heatsink (circled with the green color) are diodes which convert the AC pulses coming out the transformer back into the DC voltages. The voltages then get smoothed out by those capacitors which I circled in red and which are failing ... and then the two voltages go to the connector.
If curious, the IC330 chip in the corner is most likely a voltage reference and the potentiometer in the corner will most likely allow you to adjust the output voltage a tiny bit.
The output voltages from the transformer are compared with the reference voltage produced by that chip and if the voltage is too high or too low, a signal is sent through the optocoupler (the chip below the big transformer that sits across the cutout in the circuit board) and the controller either increases or decreases the number of pulses sent through the transformer and the voltage is adjusted where it should be.
So what you could do is use a multimeter and measure the voltage on those two capacitors I circled with red, and that's the voltages you'll need to introduce if you want to hack the board. So you'd start by cutting the traces between those two chips on the heatsink (circled in green) and the capacitors , and you'd solder wires directly on the capacitors.
You'll need to use separate dc-dc converters or linear regulators to produce 5v and 3.3v from your 12v.
I also highlighted on the display board the 3.3v linear regulator. Even though the power supply sends 5v and 3.3v to the display board, the display board most likely ignores the 3.3v and takes 5v and uses the linear regulator to produce the 3.3v it needs for the display processor.
The 5v is probably still needed.
As I said in the picture, there's also 13v in the connector, but there's a high probability you could just leave it out if the display board doesn't have an audio amplifier on it because the higher voltage is most likely for that. Even if the board has an audio amplifier, the board will still function without the audio amplifier working (no power to it)
NOW... you could simply throw out your power supply board and use one of the universal CCFL backlight boards which can function with 12v .. there's plenty of such boards on eBay, you just gotta find one that supports 4 backlight connectors.
Here's an example :
https://www.ebay.com/itm/274258962716\Here's another example :
https://www.ebay.com/itm/293056656424These boards have 4 wires :
Vin (10v...20v typically , 12v recommended) ,
GND, (ground, duh)
ADJ (adjust brightness, goes to DIM / ADJ on your cable going to the display board) and
BL / EN / ON-OFF (enable backlight - goes to display board to allow the display processor to turn off the backlight when monitor goes in stand-by / sleep)
Now all you have to do is produce 5v and 3.3v and MAYBE 13v if you really want to (but 90% sure you could just use 12v and the board won't care or not even send any voltage through that wire)
It will probably work without it. High voltages like 12v on display boards are almost always there for audio amplifiers (in case some models have speakers built in, or headphone connector).The board would function even if there is an audio amplifier on the circuit board and that amplifier gets no power through that wire.
There's dc-dc converters on eBay for a dollar or two, here's some examples :
up to 24v input , potentiometer to set output voltage :
https://www.ebay.com/itm/114513145069So you can put 12v in, and adjust potentiometer until you get 5v at the output of the board.
Use two such boards to get 5v and 3.3v for the cable going to the display board. Make sure the boards sit on something that doesn't conduct electricity, like cardboard or plastic (ex a cd case or something)
Another VERY common dc-dc converter board :
https://www.ebay.com/itm/192130689537Same story, put 12v at the input, adjust potentiometer until you measure 3.3v and 5v on the output.
Alternatively, you can make your own 5v and 3.3v using linear regulators ... get a 7805 and you have 5v from 12v ... but note you'll need to use a decent heatsink on the linear regulator as it will get hot..
For 3.3v , you can use 1117 chips or lm1085 or whatever (look in datasheet at typical example circuit and it's super easy... you can use the 5v output from the first regulator to produce 3.3v with the second regulator
So basically you could make the monitor run on 12v DC with less than 10-15$ and a few weeks of waiting for packages to come from China. Also less risky than hacking the existing power supply board.