If I were King, I would redefine the Night Watch's mission thusly:
1) Determine if the WW are all dead. We presume that the Night King created all the Others, but no one knows that for sure perhaps. Little is known about them, even the Super Tree Raven doesn't know 100% of what happened. The NW will find out.
2) North of the Wall is still not part of the Seven Kingdoms. It is still necessary to maintain a border and customs zone.
3) Whereas the Free Folk are free, they will still have conflicts over political matters. As the Wildlings re-settle the North, and as they consolidate themselves into permanent settlements, they will likely have disputes. The Nights Watch can serve as an impartial and neutral party in settling disputes and in policing the area. Like UN Peacekeepers or US Marshals. With the permission the Wildlings of course.
4) Whereas the Free Folk are free, the establishment of relations now means that the Wildlings are within the Seven Kingdoms' orbit and will become important trade partners as well as potential sources of military man-power and intelligence. It is in the 7K's interests to help develop the far North and their people. The Night's Watch can be the link between the Wildlings and 7K's and act as the administrative mechanism with which to accomplish the aforementioned policy goals.