I just bought an Electra Townie 7D, and paid an extra $100 for the equipped model: fenders, lights and a dynamo generator in the front hub to power the lights. The lead wire connects to the hub, and then goes inside the fork and comes out at the top and connects to the front light. A second wire runs through the frame from the taillight and comes out near the headlight where 2 plugs go into the headlight. However, I believe it was Tuesday when the wire that double plugs into the headlight unplugged and broke (one plug came out and the other plug stayed in but the wire broke). I noticed the hanging wire the following day and reviewed pictures from the 10th and 12th. It was connected on the 10th and not on the 12th, so my guess is it happened on the 11th . when I attached a bottle cage and balanced the bike on my leg. Maybe the handlebars and forks turned enough to cause the damage!? The shifting and brake cables also run through the frame and probably pull on the wire. The lead wire has slack to stretch when needed, but the secondary wire is snug (I plugged it back in the week before, and it is so short that it was difficult).

I’ll need to mend the broken wire, I think. It’s been 30 years since I repaired wires; back in the 80’s I’d strip wires with a razor blade, twist the copper together, and then coat with electric tape. This repair seems different; the exposed wire is silver [not copper]. If it is the same process, then twisting wires together shortens the length; however, it is not long enough all ready. Is there a better way?
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This is the headlight with one plug end of the broken wire

This is an end of the taillight wire with one plug and the other side ripped...

Clean, Strip, Shrink Tubing, Solder... Done???

Don't be afraid... You can always do it over again...

In addition to this you might consider extending the wires a bit; if it is too short now it is not going to get any longer by re-terminating it. It is hard to tell for sure but those look like what are called "Faston terminals", if you cannot salvage them. There are several widths of those kind of terminals so take one with you if you go to buy some. Perhaps Electra could provide some spares.

Your photo conveniently shows the wire size: 26 AWG. That's important if you need to buy a splice or terminal. It's also very small and can be difficult to work with. Hopefully you can salvage the existing terminal, and reinsulate it with some heat-shrink tubing as noted above. Also use some anti-oxidant paste on your connections, given the environment.

It's quite possible that the wire's open loop length wasn't enough to prevent the wire from being tugged on when the bars swung around. Or got snagged on something that pulled the wire out of the connector. Make sure the repaired wire is not too short or so long that it hangs out like a hook waiting for something to snag on. Andy