What you're experiencing is a blown inverter. This is the part of the monitor that directly controls the backlighting. The fact that the light comes on at all shows that the lights themselves work.
Finding a specific inverter can be a pain, but I've found that eBay* is your friend in these cases. I stick with "Buy it Now" and "Top Sellers" with 94-98% approval and at least 1500 transactions completed. Read the descriptions CAREFULLY and know what you're getting. Also be sure you know their return or exchange policy...almost all sellers I've seen will not accept returns, but most will accept exchanges in case of dead-on-arrival.
I deal with blown inverters on laptops all the time. Working on a G4 Titanium DVI (A1025) now with that exact problem. Found the guide right here on iFixit. I'd never worked on a Mac laptop before, so I wanted to be sure to do it right...I need not have feared, as it's easier than any other laptop inverter I've ever replaced. (The same is NOT true for most of them, apparently!)
I didn't find any manuals here for replacing inverters on LCD monitors, but I might do a couple if I get a chance. While the functional part is the same, the make, model, and procedure for doing it is a bit different for almost every monitor out there. Just Google (or Bing or Yahoo or *search engine here*) it and you'll probably find one.
I checked out eBay once I found the correct model number (931BW). There are several listings for a replacement power board. This is where the inverter is located on your monitor (as Shawn pointed out). The prices seem reasonable (the first 2 listed were $23), so it shouldn't be a problem to get it replaced, once you can get it open.
Unfortunately, I had less luck with the service manual. I found a couple that claimed to have it, but one wanted registration (didn't really want to do that) and another wanted money, but I didn't see how much. Wish I could help more on that part.
Since I wrote this, Amazon (among other places) have started offering products from third-party sellers/resellers and are fine alternatives to eBay. However, the same cautions apply. Know who you're buying from. Do your due diligence. I'm not here to promote any one source, I simply provided my own experience as a starting point. Good luck!