Well it has been more than half year since I posted, but I left the monitor aside as I got a new one, and now had a bit of time and decided to check it out more.
And since I repaired it, here is a short description in case anyone else has the same problem.
As recommended, I checked the voltages going from the power supply to the video board. The readings were:
Pins 1 and 2 - 0 V with backlight off and 2.8 and 3.2 V respectively with backlight on (presumably backlight enable)
Pin 3 - 7.18 V
Pins 4,5,6 - 5 V
Pins 7,8 - 3.3 V
Pins 9,10,11 - GND
Pins 12,13 - NC (goes to some unpopulated stuff)
Pin 14 - GND of the 12 V connector
Pin 15 - regulator enable of the 12 V connector
Well the 7 V on pin 3 were a pretty weird voltage, and absolutely unexpected so I started tracing it out.
The pin goes to the output pin of the KIA278R12PI, which is a 12 V regulator. So the output is not right at all. And on the video board, it goes via a presumably transistor into the LCD panel. Looking around a bit more nothing else seemed to be able to be a problem, and the input was decent at 12.8 V, so I decided that the regulator is probably faulty.
Thankfully the power supply had another of this regulator that was fed by the same input and was used for powering the 12 V connector. The connector was used for powering a soundbar that I don't have, so it's not used and became a donor.
Having replaced the regulator, the monitor started working properly again.