Goowie, that is somewhat odd. The circuitry for both backlights are separate. I'd try a complete reset (SMC PRAM). Make sure that the ribbon cable for the keyboard backlight is properly connected. As for your display backlight, it could be the backlight fuse as well as the backlight driver and lots of other components in between. You will have to disassemble your computer and take a look at all the connections and components. You should also check the history of the computer to ensure that there was no water damage etc. prior to you getting it. Let us know how you determined that the display works. Also, connect an external monitor and see if that works without any issues.
the backlight fuse is reference designator F9700 and it is a 3Amp 32V SMD fuse in a 0603 package. The backlight driver is reference designator U9701 and it is a LP8550 IC. Datasheet available right here.