From a couple of pages back, I think the any White Walker can reanimate a dead person, but the Night's King did his little parlor trick as a show of strength directly to Jon Snow, since he was the one who killed the WW. It was a way of saying "You killed one of mine, but look what I can do. My army is limitless."
I wouldn't even be surprised if that sort of effort took a lot out of him - sort of akin to Yoda lifting Luke's XWing out of the swamp in Empire Strikes Back - you're showing how much power you have, but it comes at a cost. If you've seen the extended edition of ROTK, they leave in the scene where Aragorn looks into the palantir they capture from Saruman to show himself to Sauron. There's nothing tactical to be gained by that move, because he can't attack Sauron there, but he's showing him "This is the face of your enemy. I'm the guy who can actually use this thing that drives everybody else nuts." Same show of force by the Night's King - those newly reanimated soldiers couldn't run out into the water (and why is that?) but it's a great way of showing Jon - who I'm sure is now enemy #1 for them - this is what I can do.
On the Valyrian steel front, I'd love to see the second or third to last episode of Game of Thrones ever include a scene where a huge homogenous army of Northerners, Southerners, and Wildlings is in the snow led by Valyrian steel wielding commanders Jon Snowgaryen, Jamie Lannister, and Brienne of Tarth; and just as the White Walkers and the Army of the Dead begin their attack, Dany, Tyrion, and you guessed it, Hodor, I mean warging Bran, fly over on the 3 dragons to take the fight to them.