While another answer has been accepted, I think a more reasonable answer is:
The detective did it.
First, note that
the text says that the detective comes the same Sunday that the husband is murdered. It does not say that the detective comes after the husband was murdered. Also note that the detective has no alibi.
when looking closer, there's no reason to assume that anyone was lying.
The cupboard was quite dirty, especially in the corners. That's why the maid was wiping them.
The wife was lazy, but the husband was annoying. And he had annoyed her so much about the dishes that she did them anyway.
The butler had his free day on Saturday, therefore the Saturday mail wasn't yet fetched when he returned on Sunday. Therefore he fetched it at that day.
Since all of them had an alibi, they did not do it. Thus only two options remain: Suicide, or the detective killing the husband.
But suicide is excluded because the text explicitly states that the husband was murdered.
So there's only one possibility remaining, which therefore has to be the correct one: The detective did it.