Is an expiatory payment (kaffarat yameen) obligatory after saying: "By Allah, you will not leave until you have prayed," and the wife leaves without praying, and how should this be handled?

You are required to offer expiation for an oath (kaffarat yamin) for your oath by Allah, in which your wife caused you to break it. The expiation is incumbent upon the one who swore the oath, not the one who caused it to be broken. You must address your wife's non-adherence to prayer, as it is a grave transgression.