A simple way to look at this is to consider what the sentence would be without the wife. You would say 'me' not 'I'. (Some say 'myself' but that sounds pretentious to me.) Then add back 'the wife' to the sentence.
If ER and Phil are the subjects of a sentence she should say "my husband and I"- e.g. "my husband and I are going to have an early night tonight".
In the article you showed, Will and Kate are the objects so it's "my wife and me"
I look at it like factor does which is correct....if the sentence in the singular uses I then....my wife and I....if the singular uses me then ...my wife and me.