I've been trying to fix some stuff very much like this lately.
I can tell you that yes you should see 12.6V on both sides of the backlight fuse. My guess is that your 11.71 is probably a multimeter error---that's what mine turned out to be on the last board.
Next, you will need to figure out on your own how to source your schematic and boardview for your particular motherboard 820-2915. Once you have those tools, you'll be able to know exactly which component to test, and where to find it on your board.
Your first stop will be to measure the voltages at the LVDS connector itself. I just glanced at your schematic and it looks like you should be seeing 3.3V at pins 2 and 3 on your connector (with display attached and laptop on).
And you should be seeing around 27V at pins 39/40 and on a nearby capacitor C9000.
On the last no backlight board I had, I saw the same 12V coming through to these pins at the LVDS. This means that the LED backlight driver chip--U9701--was not boosting. Probably because I had burned it up :) Of course if I had not seen the 12V and saw only 0V, that would mean that there was a short to ground somewhere between the connector and the source voltage. This is the kind of thinking you'll need to learn as you go on this problem.
I had to send my board out to my talented friend Louis Rossmann and he of course was able to switch out U9701 and the whole thing fixed within 90 seconds. Curses!
I think the backlight circuit on these boards is a very reasonable place to start to try and figure out how to work between the schematic and board to do some troubleshooting.
Report back with what you find as you begin measuring around U9701 and the LVDS connector. Start with voltage measurements as you are doing now, and then we can go short hunting if we need to on diode mode later.