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Before this woman can begin atonement she still has to set aside what appears to be an overwhelming narcissism. The husband and children in her letter are depicted as little planets revolving around wise mother until she makes a mistake, and then their lives are "irrevocably spoiled"? I don't think so. Life has pain, from many sources. And while this woman was incredibly selfish, her family, her ex-husband and her children, WILL get over it. Which is what she really can't stand, so that she must, must, make herself the center of attention AGAIN, by bemoaning the fate she has brought upon them all.
Better she simply find a quiet way to make amends to HER OWN FAMILY before seeking atonement through anonymous prisoners. The energy she would spend at that task would be better spent determining if she is capable of having an honest conversation with her children, which one would assume are grown, and her ex-husband about her remorse. Not in a theatrical 'all is lost' mode, but in an 'I fucked up, and I want to make amends. Can you help me understand how to do that' mode.
She could probably start by disowning her theatrics, and refusing to make her situation a writing exercise for publication in Salon.