The Night’s King of the book is a totally different character than the Night King from the show,
GRR Martin states
The Night’s King is not the first and the leader of the White Walkers.
According to the Game of Thrones wiki
GRR Martin states
in the books he is a legendary figure, akin to Lann the Clever and Brandon the Builder, and no more likely to have survived to the present day than they have.[4]
The Night’s King is not the first and the leader of the White Walkers.
According to the Game of Thrones wiki
However, the Night's King fell in love with a woman "with skin as white as the moon and eyes like blue stars"; he loved her though "her skin was cold as ice", and when he gave his seed to her he gave her his soul as well. He brought her to the Nightfort and bound the brothers of the Night's Watch to his will through sorcery. He declared himself "Night's King" and ruled over the Wall and the Gift as his own. The King in the North and Joramun, the King-Beyond-the-Wall, joined forces to defeat him.
After he was killed, it was discovered that he had been making human sacrifices to the Others - the White Walkers - and all records of him were destroyed, and uttering his name was forbidden, so it became lost to history. Nevertheless, it is believed he may have been a Bolton, a Magnar of Skagos, an Umber, a Flint, a Norrey, or a Woodfoot, though Old Nanclaims to know a version of the tale where he was actually a Stark, brother to the King-in-the-North, named Bran.