I eventually solved this problem by renaming the "Eye-Fi" folder, and creating a new "Eye-Fi" folder in its place. See below for what DIDN'T work for me.
I presume that the "C:\Documents and Settings\Jim\My Documents\My Pictures\Eye-Fi" folder got corrupted in some way, or something like that. I renamed the "Eye-Fi" folder to "Eye-Fi-bak", and I created a brand new "Eye-Fi" folder in its place. I restarted the Eye-Fi helper, which had been reporting the "Delivery Failure" error; this time there was no such error.
My backlog of photos from the last week or so (since the errors began to appear) is now downloading automatically in the background. I presume that it will download everything that was missing, but I can't confirm that just yet. Once things settle down, I intend to move each of the dated subfolders from the "Eye-Fi-bak" folder into the new "Eye-Fi" folder.
What did NOT work for me:
1. Removing and reinstalling the Eye-FI iPhone app
2. Removing and reinstalling the Eye-Fi Windows app
3. Disabling and then re-enabling the "Upload photos to..." feature in the Eye-Fi Windows app.
I hope this post helps somebody, someday, to quickly resolve this problem if it happens for them.