To start, most failures will produce a flashing power LED to indicate where the problem might be. If you have that, count the sequence and search for that online.
If 2/3 of backlight works, it is possible the failed side is affecting power for other things.
The first thing I would do is disconnect power to the backlights. Then do the flashlight test. If there is video, then the backlight needs replacement.
If there is no video, then disconnect power, remove the PSU and discharge caps where indicated. Then test as many devices on the PSU as you can. Start with diodes as they are the easiest.
It is hard to tell from the pic, but the 2 capacitors together in the top left may be expanding. The tops should be flat as the three together are.