I remembered that in some of my past laptops there was a power fault that developed over time. The cause had been to do with a drying and deteriorating solder joint inside the laptop at the DC input. And so now in this case I firstly suspected something similar was happening, but inside the PSU.
With considerable difficulty I managed to pry open the PSU case only to find that everything inside was still quite perfect. The difference was that now I had a horribly disfigured unit that needed to be taped up.
My next hunch was to investigate was the funny looking cable segment just after the PSU DC output. And behold! I found that the negative wiring had become cut underneath the insulation. I can only speculate that at some point in the past the cable was jammed in a door or a drawer, and experienced a shearing force enough to eventually cause this fault.
The temporary fix was just to cut open some of the insulation, and twist the negative wires together. Unfortunately in the process I managed to nick some of the insulation to the positive cable, but will get by for now.
No matter how you look at this, it's not pretty. However, I was able to return back to work, and also to report back what was learned.
CONCLUSION: Don't try this. Just spend $10 to $20 on a new PSU. Even get two of them, one for home and one for the bag. Spare yourself from the waste of time.
And yes, the twisted mangled result was reasonably taped up before put back into use.