Valyrian steel, fine. What we "know" about Valyrian steel is that it is apparently forged using dragonfire. Hence having the properties of dragon magic.
The issue isn't so much why VS killed the NK, but why they included the bit where he is completely invulnerable to dragonfire.
Anyway, IMO it's sloppy but not the main issue. As I said above, I can and have numerous times forgiven scenes that require too much explanation and suspension of disbelief. But for the most part they don't really matter, eg. Jaime not drowning in his armour after escaping the Dragons, the 'teleporting' characters between Dragonstone and the far north, etc etc.
My main gripe is that it's a dud payoff for the very long supernatural/3-eye-raven/white walker arc, there was no twist, no deeper meaning, and I doubt there's anywhere interesting to go now.
No one south of Winterfell has learned anything, they'll have their petty war over a chair, and live happily ever after.
Hope I'm wrong, doubt it.