Ed's theory works like this: if he just forgives her and asks nothing of her, she'll feel like she owes him. A girl of her caliber would probably see that as a trap, since a person with ill will could do her a favor and expect to call in a favor from her that she wouldn't like very much. In order to avoid that trap, he needs to ask something from her to atone. Since he wants to spend time with her and get to know her, he can kill to birds with one stone by having her help him clean.

We'll say she finds another rag or something. They can share the spray bottle. :3