Sounds like you've done the right things on the iPhone end of things, so let's look at the wireless carrier end of things.
Even if the two of you share a wireless account with AT&T, it may be that visual voicemail was assigned to your wife's phone number and not to yours. Check your wireless account and see if visual voicemail is one of the features on your phone number.
Also, here is AT&T's tutorial on setting up visual voicemail: https://www.att.com/esupport/article.html#!/wireless/KM1009152