Hi, it seems like there is an issue with the backlight circuitry on the logic board. Older MacBook Pros and A1181 MacBooks have inverters, as they use CCFL (fluorescent) backlighting for the LCD, newer computers such as your macbook air have LED lighting which don't use an inverter. Therefore, it sounds like the issue is likely with the logic board, or with the display panel.
The backlight on the LCD requires a high voltage, and therefore it is a common point of failure and corrosion when there is liquid damage. First, I would try to find any burned/corroded areas on the logic board, especially around the LCD connector, and clean with a q-tip and isopropanol.
If that doesn't solve the issue, I would take it to a reputable 3rd party MacBook repair specialist that does component-level motherboard repair as they will be able to fix your laptop for a lot less then what Apple charges. Find someone who does a free diagnosis, as there is a small chance that display panel needs replacement. You can also take it to Apple of course, but they will charge much more as they will simply replace the whole logic board or whole logic board assembly.