If the trackpad connector latch on the logic board has broken off, this is the cause of the issue. When the trackpad is not fully recognised, the keyboard will not work (other than the backlight and power button).
You have 2 options;
1. Solder a new trackpad connector; AFAIK you can't buy these new, so you will have to desolder one from a donor board (hard to come by for this model, unless you regularly repair MacBooks).
2. Add tape (or something thin) on the end of the trackpad cable, in hopes that this will put enough pressure on the cable for the trackpad to be recognised. Not a very good fix and possibly not a long lasting one, but it should do the job.