PS To fix the broken wire, I took the pin from out of the connector and (rather than try to uncrimp the wire) I just soldered another wire about 1.5" long onto the connector.
I then stripped about 5mm off the end of that wire and also 5mm off the end of the wire from the loom and crimped a spade connector and its counterpart onto the ends before wrapping the whole thing with electrical tape.
If I'd had some dielectric grease, I'd have added some of that to the new connections and into the plug connectors to keep them from getting corroded again.
But that's for another day...!
I then stripped about 5mm off the end of that wire and also 5mm off the end of the wire from the loom and crimped a spade connector and its counterpart onto the ends before wrapping the whole thing with electrical tape.
If I'd had some dielectric grease, I'd have added some of that to the new connections and into the plug connectors to keep them from getting corroded again.
But that's for another day...!