Further to our saga, and I hope concluding it, Trend Micro have been following up our IE9 problem and applying various fixes. The last one was what they called an ATTK Clean Kit, which ran a big scan, found 5 items of concern and got 'fixed' them.
Now IE runs well again, and I think our computer, if it by some chance happened to possess a degree of sentience, would feel like it's been gone through with a wire-brush pull through followed by a dose of salts.
Thanks to everyone, especially Milo, for your help.
Also a thumbs up to Trend Micro for customer service - admittedly their software didn't stop the infection in the first place, but their support team (via the charmingly named Rain) kept following up until the problem was solved, unlike many companies of various kinds we've dealt with over the years.