Let us know the board number, which should be on the bottom right, since I do not think this is the 2012 version, most likely 2013 or 2014. Get a board view and schematic file for the board number, to find out the value of the components, or let me know the number and I will check.
You cannot resolder the burnt component (ferrite bead), that will need replacing. This part usually burnt when the LCD cable is chipped when connecting, which will also melt the inside of the LCD connector too (so would recommend replacing this too). You can shine a light inside the LCD connector and see if it looks slightly off. When this component is not working properly, you will not get any picture at all (not just no backlight).
Replace the backlight fuse for good measure too (part with the P on it). Looks a bit green around the ferrite bead, which would indicate liquid, but I am not sure if that is just a camera issue.