I don't know anything about the Kilma relay because I have never had to mess with one, but I can tell you something about your compressor not kicking in.

The most likely reason that the compressor will not kick in would be insufficient refrigerant in the system. There is a pressure sensor that prevents the compressor from kicking in if there is insufficient refrigerant. This serves to prevent compressor damage.

Hook up gauges and see if you have static pressure, which should be around 80 PSI. If you do not, then the system is low on refrigerant and that's the road you need to follow.

Good luck,