According to iFixit (and every other source I could find), what you probably need is called an LED Driver board.
The backlighting of LED-lit displays is a bit different than standard CCFL-lit LCD displays. For one, there is no "inverter". I found this out the hard way working on a Sony several months ago. It was my first foray into LED displays on laptops (or anywhere, to be honest).
At an educated guess, I'd bet the LED Driver board controls the keyboard lighting as well.
Replacing the LCD panel will not fix your problem, as it has no control over the lighting, regardless of what type is used.
There is a guide here on iFixit that will show you how to do this. You will need this part.
I hope this helps!
It seems I need to learn to read the questions better. I missed the "unibody" reference. I'll assume this is, in fact, the model in question. (I can't actually edit the device listed on the page...rep not high enough, I guess.)
That said, this may be the correct one.
Unfortunately, this is the entire assembly, rather than just a part.
I checked eBay and came up with this, which may be more appropriate, depending on how much you feel like opening up that display assembly.
I used the search terms "macbook unibody 15 led" on eBay to get that. As usual, I only looked at "Top Sellers" under "Buy It Now".
Let us know if we're on the right track! Otherwise, I'll wait for more detailed information about the system.