The Night King is a Stark, and is periodically replaced by another Stark in an ancient agreement - the Prince who was promised, from the family of the Kings of Winter. Night King is coming for Bran, his replacement, who has the necessary warging qualifications.
Times have changed though, and modern surveillance techniques result in him getting busted while stalking his prey.
Jon Snow is a Stark and a Targaryen. I figure the two at least get into combat with each other. Jon , already being ‘undead’ means the night King has no power over his dead body.