Plan A: Find a short length of very fine uninsulated wire. A single
strand from some stranded wire is what I use. Wrap it around the
insulation about 2 times, and then continue wrapping around the tinsel
wire. Clip off the excess at the end. Solder the wrapped wire to the
replacement connector.
Jeff L.

Problem is that it will break almost instantly at the transition point from
stiff to flexible.

The technique mentioned above seems to be an excellent solution to that
problem.
Mike

I don't understand the technique.

The wire is enameled (insulated). Do I prepare the wire by burning (or
sanding) off some of the enamel first?

I start wrapping back a way and wrap toward the end of the wire?

Then I solder not the tinsel wire but only the wrapping wire beyond the end
of the tinsel wire? This looks like there is no actual soldering of the
tinsel at all (which is intentional, I presume, to avoid stress points).