ah yes Tinsel wire.
joy
What you need to do is get a fairly fine strand of copper wire, like 32 gauge ( single strand from some 18 ar smaller stranded wire ) . then wrap this as tight as you can around a section of the wire, wrapping the string, foils of copper and all. Once this is wrapped for say 1/4" or so, solder it.
You will end up with something you can then solder to the phone plug. You need to be quick with the soldering iron, so you get ellectrical connection between the foils and the wrap before you melt/burn away all the strings.
The wire will need to be supported where it leaves the plug since the foils will easily break soldered wrap begins. Folde the braided sheild back over the cables outer jacket, and crimp to the plug with the cable restraint that is ususally used to on the cable alone.
Sometimes it is best to secure the wire to the plug first so there is little movement, stressing the connection, and breaking the foils.
Very tedious , but it will work. Normally this wire was terminated with little pins that had teeth inside, that bit through the insulation. The pins could then be terminated however you needed. Common in phone cords, etc where highly flexible but light cords are required.
Good luck!