In "The Death of Ivan Ilych," how does Ivan come to achieve redemption in spite of his meaningless life?
It is clear that what allows Ivan Illyich to achieve redemption, if indeed redemption is what he achieves, is a recognition of how he has wasted his life by focusing on the material alone. His realisation of this allows him to feel sorry for his wife and son, which in turn gives him the peace that has eluded him and the courage he needs to face death head on. Note how after his attempt to utter the words "I forgive you" to his wife and son, he experiences a sudden peace and freedom:
And suddenly it grew clear to him that what had been oppressing him and would not leave him was all dropping away at once from two sides, from ten sides, and from all sides. He was sorry for them, he must act so as not to hurt them: release them and free himself from these sufferings. "How good and how simple!" he thought. "And the pain?" he asked himself. "What has become of it? Where are you, pain?"
Note the way that his act of forgiving his wife and son gives him the calm and acceptance of what is to come that he has been unable to attain up until this point. His realisation of how he has wasted his life, and his determination to do something about it while he can by forgiving and apologising, allows him to achieve the redemption that your question refers to.