So, what you're saying is that the only reason for this ground is for the touch-sense?
If this is true, I have other questions. The braided covering on the wires, is it stainless steel or nickel or silver plated copper? If it is nickel or silver plated copper, then solder would be fine to secure it further, as I am of the opinion that wrapping it is inadequate. However, if it is stainless steel, I am not sure how easy it is to solder, although I have been successful at soldering stainless steel, it is difficult. You could be more successful if you wrap the single wire in a longer spiral and then use ty-wraps (multiple) to further secure it and then use the heat shrink over the ty-wraps.
Is the single wire copper with nickel or silver plating or is it aluminum? If it is aluminum, then I recommend the ty-wrap/heat shrink method as opposed to soldering.
Hope this helps...