@HWTech, many thanks for your helpful reply.
Both the 60 W and 85 W chargers are the T type. I've wiggled the 85 W until I can't wiggle any more. I'm sure it's mating properly. The 60 W connects and lights up immediately with no wiggling required. And of course this is now the second 85 W charger I've tried. I've been using Macbooks with Magsafe chargers for a long time, and always keep the charging port scrupulously clean. I'm certain that the physical connection between the charger and the port isn't the issue.
Obviously it could be a DC-In board or logic board problem, in which case I'm not sure how to proceed as the laptop is long out of warranty and a repair such as that is sure to be uneconomic. Perhaps I'm left with using the 60 W charger and keeping a close eye on it in case it overheats.
Here's the weirdest thing though: just one time, while I was running hardware diagnostics as suggested by leroydouglas above, I plugged the 85 W charger in and it worked. Just once. It was after I'd run the test for the first time with no adaptor connected, and it told me to plug it in. I connected the 85 W and the light came on. Ran diagnostics again and it happily recognised the 85 W charger. All was apparently well. Leaving it connected, I started the laptop, let the OS load, then opened the system report. The charger light stayed on the whole time, the status bar confirmed the battery was charging, and the system report confirmed the 85 W charger was connected. Everything normal. Bravely I unplugged the adaptor, and immediately reconnected it.
Nothing.
That was about half an hour ago. Since then I've plugged and unplugged, connected and reconnected, switched power cables, powered off and on again, run diagnostics with each adaptor and with none. Back where I started – the 60 W wrks fine, but not a blink or a flicker from the 85 W.
Thanks so much for trying to help. This is doing my head in.