She didn't actually surprise the Night King. We see him slowly drawing his sword, see her leap towards him and he suddenly spins and grabs her by the throat. The way he does this is very deliberate. He appears to be expecting her attack and deliberately appearing unaware. (If you watch the scene very carefully, one thing you will see is that his eyes do not move at all when he changes from pulling his sword to spinning towards her attack.)
Notice that the other white walker also doesn't appear to have any expression as he turns. He appears to be dispassionately observing something. There's no surprise on his face.
The wights stayed silent not because they didn't see her, but because the Night King was keeping them silent. This was a ceremonial killing, not an attack.
This is not the first time in the battle the Night King showed such indifference and arrogance. He does this twice before in this episode alone, once just before Daenerys unleashes dragon fire, another time as Jon Snow starts to charge. In the latter case, he could have raised the dead long before. He's rubbing Jon Snow's face in his failure.
This is an arrogant creature who is amusing himself by toying with humans, rubbing their faces in the powerlessness of their attacks. That's exactly how he reacts with Arya. He let's her think she's got a shot, then spins around to demonstrate his superiority.
Then, she kills him with trickery.