Active Directory name changes

I have a user that had a name change and I have gotten everything changed except the actual network account.   For examle, she had been mjones on my domain.  Now she is msmith.   I thought it should be as easy and going into her mjones properties and just changing her name.     NOT A GOOD IDEA.  So I created a msmith user account.  Now when doing things, there are times that her old and new account seem to have "blended".  I dont know how to explain it.   For example.   She logs in as msmith   But when going to My computer, her home drive says mjones

I don't want to make matters worse so what would have been my proper way to keep her profile and settings and get her name changed.   Should I have used the copy function in active directory and created it that way?