Why no conductivity with this wire?
There is no conductivity even if I place the probes on the same length a short distance apart.


That is likely because of clear insulating "lacquer" on the copper wire strands, which is difficult to see visually. You need to scrape those wire strands down to the bare wire (e.g. with fine sandpaper or a sharp blade) to make electrical contact.


  How do I solder the wires to a new connector?


Carefully! Some of those insulating "lacquer" coatings will burn off under the heat of a soldering iron, allowing soldering to be successful. However:


sometimes extra flux is needed;
sometimes even with extra flux, the lacquer has a high melting point and has to be removed mechanically (e.g. sandpaper) first;
sometimes the PVC insulation will melt very quickly when trying to solder the wire.