Justin--this is an easy problem, very solvable.
1.) a hard reset will cure many backlight problems in the iPad 2--I'm sure you've tried that
2.) it is never the coil, I swear I'm going to put that on a T-shirt!
3.) it is always the filter, and sometimes also the diode.
Your filter is located under that EMI shield bracket near the diode. The filter is the small component that looks black--I'm on a phone so I can't label it for you. If it is burned you can see visual damage. You'll need to replace the filter. If you're not a microsolderer then send the board to someone like me or Marcus's (cyberdoc) new technician since he's not doing repair anymore, or any of the eBay folks. It is very straightforward.
Now measure your diode. Set your multimeter to diode testing and put the black lead on one leg of the diode, red lead on the other--should get a reading of 1 or 150-180. Switch leads and take second reading. If you have a 1 in both directions diode is bad, otherwise diode is good (usually it is good)
With a new filter, good diode and good coil--the only variables left are the lcd connector and the lcd itself
Jessa