The only way that he's going to do anything is if you tell him (or write it out). He definitely should be responsible for the dishes, the picking up or even for setting up and running activities here and there. He shouldnt' get to sit and interact with the kids while you're cleaning.
Just tell him what to do. He's probably wondering why you're not asking him to do these things anyways.
Just like with the kiddos we set expectations for our assistants. If we aren't clear about our expectations for behavior they have to guess and that's not fair to you or him.