[imnuts]

By default, the keyboard backlight uses the light sensor to determine how bright to light up and/or be on at all. Not sure where the sensor is, but it is quite easy to cover it up with your hand while typing. When I use my keyboard, the backlight is constantly changing. Try this cab which will disable the keyboard light sensor to see if the backlight stays on for you. After installing, soft reset your phone and the keyboard backlight should now be either on or off.

EDIT: That could work too ^^^