For those who have read the books,

I'll let you know if/when he ever finishes them! (Not looking likely at this point.)

I've read all of the released books, but not any of the pre-release Winds of Winter chapters. I don't recall anything that addresses your question. Maybe someone here with better knowledge of the lore caught something I missed.

The Night King isn't in the books (yet). Just the odd Walker/Other

Bran gave us a bit of an answer tonight, although I’m not sure that there isn’t something more to be revealed before the end.

There's no mention of the Night King at all in the books. The Others (what the White Walkers are called in the books) are very shadowy and mysterious and there's been very little explanation of what they are, and absolutely none on what they want (though it has been hinted that there may be more to them than meets the eye). I think it's expected that the next book, The Winds of Winter, will contain some revelations about them. It may be that there is a Night King. But we don't know yet.

Although, in the books, there is a mention of a historical figure called Night's King, who was actually a member of the Night's Watch who took over one of the castles along the Wall, married a mysterious woman with blue eyes (it's thought by some that she may have been an Other), bewitched the men under his command, and ruled his castle as a king making sacrifices to the Others until he was brought down by an alliance of a king of Winterfell and a king beyond the wall.

Whether that has any relevance to what's going on now is unknown.

if you want some mind-blowing theories on the Night King, you should check out one of the Youtube channels on it, like Alt Shift X. One of the theories is that the Knight King is
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Potential spoilers ahead
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Brandon Stark (that went back in time and got stuck)