Y.M.
Maybe I'm out of touch but I don't understand how telling an 11 year old to do her chores is a "mommy thing," forcing a relationship or being disrespectful. Don't you think it would be best if your children did follow simple household rules, such as do your chores, whether they are told to do so by their father or any other adult who is present and involved in the situation?
You don't really have to 'deal.' Your daughter does. Tell her that it is okay for her to like or dislike this woman but that either way she needs to be respectful. Find out what her true issues with this woman are (does she feel like dad doesn't pay attention to her if girlfriend is around? Is she hoping that you and dad will get back together some day?) and address those.
Reassure your daughter that you and dad love her and that you will always be mom and he will always be dad regardless of whatever adult relationships either one of you bring to the table.
Best of luck.