Safari not accepting website certificate
I've since solved the problem, though I'm not sure if this is the right answer for everyone.

The problem seemed to come from the certificate automatically generated from apple (com.apple.idms.appleid.prd.[large character string]). A quick Google search didn't tell me what this certificate was for, but I'm sure that deleting it was probably not the best solution. Ignoring that thought, I deleted it anyway and now I can log in without a problem. I'm sure that something bad will happen with other things on my computer (private key, other certificates, something), but it hasn't happened yet so I'm calling it a win.

UPDATE:

A couple of months on, I still haven't had anything bad happen so I guess this solution didn't break anything else.

UPDATE 2:

About a year on, this problem still comes up from time to time (even with 10.10). I keep deleting the certificate in the keychain and it keeps working for a while.