If it is a rubber dome keyboard, then it works most likely by having metallic pads on flimsy plastic membranes. Re-routing traces on a membrane is very difficult and is not very durable. I have tried that, and destroyed a pricey keyboard in the process ...
You can't solder on a membrane, but .. depending on the controller, perhaps you could solder the two wires to the controller's circuit board (if it even has one) without the soldering points getting in the way of the membrane contacts.
As Lanx suggested, I too suggest that you get a mechanical switch, such as the popular Cherry MX. The Cherry MX switch is the same type (family) as in your Kinesis keyboard, You will need a Cherry-compatible key cap.
A Cherry MX switch can be mounted in a square hole of the right dimension on a plate of metal or plastic. It may be tempting to cut just a hole in the keyboard's case, but that will not give you the correct height.
Alternatively, you could get a Cherry MX switch that is soldered to a piece of printed circuit board cut from an old Cherry keyboard. There are several members here who have modified their Cherry keyboards by cutting the numeric keypad, and one of them may be able to provide you with their leftovers.