I finally remembered to again research the problem with my primary notebook and found this: Report Compaq Presario CQ60 won’t turn on [Solved/Closed].
And was smacked in the face by this:
1. Remove the battery from the computer.
2. Turn on the computer using the cord. Remember not to put back the battery yet. Just the cable and the computer connected. If the computer turns on, then, you need to replace the battery.
3. If it still not turning on by using the cord alone. Try to wiggle the cord to check whether it’s loose or not. Check if there are some signs of power on the computer. If there is a power like lights lit in the computer, then, it’s the computer that’s having the problem. You need to bring it to a technician if that’s the situation.
Boldfaced emphasis added by me.
Wait. What? It can run without the battery? It never has!
So I did the cord wiggling bit.
And BAM! Lights!
Something is wrong in the cord leading to the connector or the connector itself.
But the computer is back! I’m typing this on it.
One source of misery (mostly) gone.
Tomorrow I can get back to being more productive (the day is wrapping up here).
If the Wiggle Maneuver still works …
Previously here:
Mike Cane's Atomic Supermen 
Try Sugru, a mouldable silicone rubber putty. You can wiggle the connector into place and when it is working place the sugru around it to maintain position once it cures. Really neat stuff
https://sugru.com
I’ve heard of that. Thanks for the reminder.
Back in the 90s there was a range of Apple laptops with a faulty power connector and the fix was to solder the connector to the board which had minimal solder. An electronics repair shop might be able to help.
Thanks for the tip.