I think the problem may stem from that I renamed my account. When Club Nintendo launched, I already had a MyNintendo account for reasons I do not recall, and so I just converted that instead of creating a new one. But its username and password were still the randomly generated mass of letters and numbers from when it was created, so I changed it to something comprehensible. I also tried entering in the old account name and password, as well as trying the old name with the new password, but no luck.
I'll soon email Nintendo, so we'll see if they can offer a solution.