The connector between the panel and the motherboard might need re-seating, or the lamp is dying. If it’s an LED backlight, they don’t require the high voltage CCFL based ones did, so that rules out a faulty inverter.
It wouldn’t be that difficult to dismantle it and take a look, you just have to go slowly and examine the connections to determine whether the locks need to be lifted/tilted etc to prevent damage.
I had a couple of BenQ monitors which I tossed after a year because the LED arrays would randomly not switch on, meaning the middle or one of the outer edges would be too dim to read. I didn’t bother trying to fix them as they were cheap, I just parted them out to the guys at work who teach electronics.
The battery not charging is likely just a dead battery - not all batteries last as long as they are meant to. The charge circuit could be broken, but meh. If the power supply as a whole was broken I’d expect you’d see far more problems related to performance and reliability (in terms of crashing applications, etc.)
Back up any data which is important before you go inside it with any tools.