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.