First of all - I know opinions vary on this, but I would not feel comfortable leaving a 9 year old and an 11 year old home alone. Just for example, do you think either of them could successfully perform the Heimlich maneuver on the other? I would spring for the sitter for all 5 days. Working from home is not easy with interruptions from kids.
Second - as to your questions about whether to have him leave or not - it depends on a lot of factors. Is an affair a dealbreaker for you? Are you able to use him still being in the house to showcase the "new you" ? Does it financially benefit you for him to stay? You may want to make a Pros and Cons list and sit on it for a while.
If you're pretty definite that an affair is a deal-breaker for you, then you already know the days and times when it is probably happening, and could hire someone to tail him or (although this may not be legal in your state) put a tracking device on his car.
One question - does the current babysitter say he stays working in the house while she's there, or does he leave when she is there?
