I forgot to take a photo of the first step: take a sharp razor blade and carefully remove the insulation along each side of the twinlead for about 3-4" so that you can pull the wires out. Twist each wire so that the strands are together as in a normal stranded wire. Fold the now-wireless insulation back on itself, leaving space at the bend for a little loop on top.
Put a small zip tie through this not-yet-secured loop and zip it to form another small loop. This loop will be used as an attachment for a support line, and to secure the antenna wires.
Take two insulated wires that are of identical length (length not critical, my wires were originally 44' long and now a bit shorter after the rebuild), strip about 5/8" from one end of each, and use an overhand knot to tie the stripped end of each wire to the zip tie. Be sure to leave a few inches beyond the knot. The goal is for the beyond-the-knot segments of the wire to not receive any tension, no matter how hard you pull on either of the wires. The wires from the twinlead will wrap around the stripped ends (but not yet). The wire size isn't critical either, I used 24-gauge stranded hookup wire as that is what I had on hand.
Add a second zip tie to secure the loop in the twinlead. Be sure to run the antenna wires through the zip tie as shown. Cinch the zip tie down tight and trim off the end.
On each side of the twinlead, twist the twinlead wires around the stripped ends of the antenna wires. Apply solder to the ends of these splices but don't allow solder to flow all the way up to the tops of the splices. The idea is to leave the top flexible, just in case tension ends up being applied to the splice (which should never happen, but best to be safe).
You'd think (or at least I thought) that a single overhand knot would not be sufficient to keep the wire in place and not slip. However, the "corrugations" on the inside of the zip tie bite into the wires' insulation and hold them in place.I did some destructive pull tests and discovered that the 24-AWG wire will actually break at some other point in the wire before the knot gives way.