As I said in the piece on the Stormlands, my idea would be that the Durrandons would be “Andalized” – a puppet king would be forced to marry the daughter of an Andal warlord who would rule in the king’s name until the king’s son, who would very definitely be raised in the Faith of the Seven and speaking the Andal language, would take over and cement the new order of things.
This would mean that the Andals would have truly conquered the Vale, the Riverlands, the Stormlands, the Iron Islands (sort of), and Dorne (at least in parts), while being assimilated into the more functional states of the Westerlands and the Reach. This would leave the North as the lone kingdom that never fell or intermarried into the Andals.