Living in chicago my whole life, there are really only two possible options outside the American club.
Lake Geniva is a beautiful town with manY great rental opportunities. My family has a house at geniva national and we have loved the service and amenities there. You can find some nice properies there that are very kid friendly. It is where most wealthy Chicago families weekend in the summer (think much smaller hamptons).
The other is eagle ridge in galena (on the missippi). While it is by no means luxury, they have some great houses to rent with unbelievable Midwestern scenery. They have good golf, nice trails, a quait town and winer. I can give more detail if you would like.