Well, I now know what the problem is but, for the life of me, I can't figure out why it would have happened. What burns me up is the amount of time that it takes to research and try to remedy an issue since Microsoft does not make it easy to find answers. After a few false starts and a another try at Fix-it (all that did was say that I had a problem that it couldn't fix) I did a keyword search with the specific error code for all of the failed updates: 0x80246008. I found that not only was Background Intelligent Transfer Services (BITS) not enabled it was also missing entirely from Services!
According to Microsoft Article ID: 883614 I could restore BITS by the run command given in the article. However, when running the command it asked for the disc with XP SP3 and my husband's disc was an earlier version. I then followed the advice to go into the Windows directory where the Service Pack i386 files are and that seemed to work, meaning that BITS now appeared in Services. The service said it was stopped. When I clicked on Start I got an error message "Could not start the Background Intelligent Transfer Services service on Local Computer. Error2: The system cannot find the file specified." The path to the executable is listed as C:\WINDOWS\system32\svchost.exe -k netsvcs.
That's as far as I got this evening before I ran out of time and decided to update you on the issue so far.
Since the file is missing I can think of two scenarios:
1. malware has gotten onto my husband's computer despite his scans (listed below), or
2. in his zeal to keep malware off his computer, my husband has run Malwarebytes and Superantispyware standalone scans in addition to our Vipre antivirus scan. Perhaps the file was accidentally quarantined or deleted. (He does not have three antimalware programs running at once, just Vipre with the other two as second and third opinions.)
I will have to check the logs on the two secondary scanners to see if anything is listed as quarantined. If I don't see anything I wonder if going back to a System Restore point in the October security update week might fix everything. That was the last time that Microsoft Updates worked correctly.
Also, I wonder why sfc /scannow did not find and replace the missing BITS in its scan. Likewise for the svchost file?
As for keeping IE patched I agree. Windows XP only is capable of IE 8 and neither my husband nor I use it when on an XP computer. We use Firefox both on XP and on my Windows 7 computer.