I ran DISM exactly as the article instructed and it reported success. I then ran Windows Update and got the same failures as I had previously witnessed, with the same "We couldn't complete the update. Undoing changes," etc...
The updates that failed (and have consistently failed several times) were:
KB3000061
KB2989542
KB2978041
KB2987107
KB3000988
Previously, these updates also failed:
KB2998174
KB2972103
KB3001237
KB3000869
KB2977292
KB2968296
Sorry I didn't provide the label on the updates, but I couldn't find a way to print out the log.