How to complete this circuit
Small colored wires like this have an enamel coating that insulates the wire so that it does not short out to other wires or metal objects/surfaces. It is this coating that is making it so that you cannot just twist the wires together to complete the circuit.

The conventional way to rejoin wires of this ilk is to twist and solder. But first it will be necessary to very carefully scrape enamel coating off the surface of the wire using a sharp utility knife blade or safety razor blade.