Well, unfortunately the mint died - I think it was already on its last legs from the insect attack, and going from warm to cold finished the job. However, it would have helped, I suspect, had I actually known what I was talking about to get advice!!! It wasn't white fly (which I've now experienced outside). I think it must have been red spider mite, which I expected to be red. And these where white. But it's the damage that they do which is red, right? These were microscopic white insect thingies. You couldn't see the wings.
And when I put a parsley plant in the same place on the kitchen windowsill quite a while later, the blighters came back! I washed them under the tap, taking rusty lady's advice. That helped, in so far as it reduced them, but didn't get rid of them.
I daren't have herbs in the kitchen anymore, but seem to be able to have cut and come again lettuce in a trough on the windowsill without the dreaded Whatever-They-Are's attacking.