Update: After scanning the drive, running a virus checker and all the other recommended steps (apart for a power reboot - the netbook has an internal battery), I ran the repair for Windows Updater in Safe-networking mode, twice.
Once I'd rebooted, I ran windows update. It announced that it needed to "Install New Windows update software" I gave it the go-ahead - and the progress bar ran for about ten minutes. Windows updater then closed. I ran it again, and got the same message.
I OK'd that also, and it returned error 8024001B (which I later learned is, though scary, just a conflict between updates). I tried again, as it suggested, and WU asked me to restart, which it did, with the usual progress counter. On reboot, my update history showed that "Windows Update Agent 7.8.9200.16924" had been installed. Yay!
So thank you for a tool that's got me out of an awful fix.
I've now found 160+(!) updates pending, and in the process of downloading them. Fingers crossed!
[Not marked as resolved, as there's a pending question about parse points]