How has Edmund redeemed himself by the end of the story?

He does not redeem himself. It is Aslan who redeems him (by dying).

Edmund can only accept redemption by being sorry for what he did and living in the consciousness that he does not deserve to be king.

However, as King of Narnia, he shows himself just and perceptive. In the second book, he is the first to believe Lucy when she says she has seen Aslan.