Quoted from Agent_Hero:They could've had Jon kill one earlier in the episode? I don't know, but somehow she should've been able to get close thanks to her assassin skills rather than just going Crouching Tiger on him.
The only way to kill the Walkers was with Valerian Steel or Dragon Glass and when stabbed they shatter into cubes of ice (as we already knew from Sam killing one). Pretty sure Arya's assassin skills do not allow her to wear a face of ice cubes...