The answer to this question can be found in Chapter Sixteen, when Aslan leads the army of new recruits he has just gained from the palace of the White Witch and taken them to join the army of Peter and Edmund against the forces of the White Witch. The arrival of this new force, fresh for battle and ready to gain vengeance against the White Witch who has imprisoned them for so long, is obviously a boost to the morale of the Narnians who have already engaged the forces of evil, and the end of the fight is quick in coming.
However, to answer your question, it is Aslan himself who kills the witch. Note what the text tells us:
Then with a roar that shook all Narnia from the western lamp-post to the shores of the eastern sea the great beast flung himself upon the White Witch. Lucy saw her face lifted towards him for one second with an expression of terror and amazement. The Lion and Witch had rolled over together but with the Witch underneath; and at the same moment all war-like creatures whom Aslan had led from the Witch's house rushed madly on the enemy lines...
Thus the White Witch is killed by Aslan and her armies vanquished and defeated.