Thank **** Arya is out of there.
He told her 'either way a face will be added to the wall' by implication, if she didn't complete the kill, then she would be added to the wall. He has now dispatched sidekick to go kill her. He likes Arya, he wants to set her free, he also knows she won't make it as a faceless person.
So he sent rage girl to kill her, not himself.
Arya was trained to fight in blindness, 'the dark' of blowing out the candle, rage girl will find her, Arya will win, and rage girls face will be added to the wall. Thus a face added to the wall, Arya free, and all the words of a payment and a death fulfilled.
He sent rage girl, not himself, as he would kill her easily, he feels Arya has a chance against rage girl.
-edit, I still do wonder just what the faceless man was doing in Westeros in the first place, as he obviously allowed himself to be captured, and dumped into the nights watch (maybe to escape later) but why would he be there, and why the complete fascination with Arya. As now she knows the value of life, the value of not killing, perhaps that was his final lesson to her.