Dawn’s Justice Ends the Long Night: The Night’s King in the Black Gate

[Sly Wren]

A potential addendum to the topic: the idea that the Night's King must be killed by the Day's King to restore order and the life cycle.

In Game, Jon VII, the same chapter where Jon and Ghost help find the dead rangers, and the chapter where Jon kills the wight--in that chapter, as they are coming back to the Wall:

The day was grey, damp, overcast, the sort of day that made you wish for rain. No wind stirred the wood; the air hung humid and heavy, and Jon's clothes clung to his skin. It was warm. Too warm. The Wall was weeping copiously, had been weeping for days, and sometimes Jon even imagined it was shrinking.

The old men called this weather spirit summer, and said it meant the season was giving up its ghosts at last. After this the cold would come, they warned, and a long summer always meant a long winter. This summer had lasted ten years. Jon had been a babe in arms when it began. 

Ghost ran with them for a time and then vanished among the trees. Without the direwolf, Jon felt almost naked. He found himself glancing at every shadow with unease. Unbidden, he thought back on the tales that Old Nan used to tell them, when he was a boy at Winterfell. He could almost hear her voice again, and the click-click-click of her needles. In that darkness, the Others came riding, she used to say, dropping her voice lower and lower. Cold and dead they were, and they hated iron and fire and the touch of the sun, and every living creature with hot blood in its veins. Holdfasts and cities and kingdoms of men all fell before them, as they moved south on pale dead horses, leading hosts of the slain. They fed their dead servants on the flesh of human children … Game, Jon VII

Basically, the phrasing the old men use to describe the Wall fits with the stories of Old Nan. The life cycle--giving up the ghosts of summer, to bring the ghosts of winter.

Not exactly Day's King and Night's King--but Summer and Winter and their ghosts. As Jon needs his Ghost. Jon's learning part of what he will need to know to bring back the day.