Hi Adrian, I have previously used wire from old headphones/earbuds, the center wire in those usually works fine for grounding the arm, or if you have some of the original (grey or white) signal cable, a piece of the centre wire from that will also be good for the grounding.
So as I understand, the signal wires are intact, the grounding from the arm is intakt on the first leg of the "journey" ie from underneath the arm to the underside of the arm pillar. Then it is a case of soldering a piece of wire to the small bracket on the underside and you're done.
The little yellow loop should be soldered to the underside of the arm itself on one end, and to the bracket on the pillar screw on the other (green arrow). To complete you'll need a wire to substitute the other yellow wire (red arrow):
cheers
henrik
ps: make sure you have enough length in your wires, otherwise the arm will be tethered and not move freely. Also, do not put the relay immediately underneath the arm pillar, it will not be enough space.