I'm not sure that Amir believes any actions will totally atone for his past sins against Hassan. Even at the end of the novel, when Amir makes a breakthrough with Sohrab,
It didn't make everything all right. It didn't make anything all right. (Chapter 25)
But Amir is willing to risk his life to bring back his nephew from Taliban-controlled Afghanistan, and he successfully accomplishes his goal even if he nevers finds complete redemption. Amir believes Rahim Khan is correct when he is told that
There is a way to be good again...A way to end the cycle. (Chapter 18)
As Amir takes a beating from Assef in his old home as he bargains for Sohrab's freedom, Amir celebrates the pain of redemption.
My body was broken--just how badly I wouldn't find out until later--but I felt healed. Healed at last. (Chapter Twenty-Two)
Amir reaches out to Allah in prayer after Sohrab's suicide attempt, praying that "my sins have not caught with me the way I'd always feared they would." But in the end, "a small, wondrous thing happened" when Amir and Soraya take Sohrab to a park in Fremont. Sohrab shows interest in the kite that Amir has purchased, and the boy assists Amir in cutting the last kite. Sorab smiles. "Lopsided... Hardly there. But there." And Sohrab nods when Amir asks if he can run the kite for him.
"For you, a thousand times over," I heard myself say.
... It was only a smile, nothing more...
But I'll take it. With open arms. Because when spring comes, it melts the snow one flake at a time, and maybe I just witnessed the first flake melting. (Chapter Twenty-Five)