Were there any storms right before you found the monitor off? It's possible you either got hit by a surge or you temporarily lost power. Your computer can be configured to start up automatically after losing power, but your monitor may not turn back on automatically, which would explain it being off. A surge could have damaged any number of components of the computer, including the monitor.
Nobody has mentioned reinstalling the drivers, yet. You should try that first. It's entirely possible they got corrupted somehow, perhaps by a recent update. Granted, I don't think that would cause your monitor to lose its settings, it's still a good first step.
Once you've determined it's not the drivers, the only real way to test this is with spare hardware of some sort. I saw you don't have any spare hardware, but if you can borrow from someone else, it will help a lot. It does sound suspiciously like the monitor going bad, but I don't know any real way to verify it's definitely the monitor without extra equipment.
Once you get spare hardware:
- Replace the monitor. If the replacement monitor works fine, you know it's the old monitor. If it's still under warranty, contact LG. If not, it's time for a new one.
- Try the current monitor on a different computer. If the monitor works fine on a different computer, you know it's something in the original one.
- If you've determined it's definitely the computer, you'll need to troubleshoot those components one by one, starting with the video card.
If you don't have a friend with spare hardware, you might check your area for computer clubs. You can also check Craigslist for cheap spare parts in your area. They don't have to be good, they just have to be functional.