in "Two Kinds" how does the narrator's mother show forgiveness?
The end of "Two Kinds" represents both the end of the conflict between Jing Mei and her mother that can be traced throughout this short story and Jing Mei's own self-acceptance of herself as an individual.
Before her mother dies, Jing Mei is given the piano by her mother. It is interesting that she describes this as a "shiny trophy" - a metaphor that clearly indicates her feelings about the piano and about her conflict with her mother over her piano playing. Jing Mei regards the piano as a "shiny trophy" because she has won it, but on her own terms, rather than through being forced to do something by her mother. We can also see this as an expression of forgiveness on the part of the mother - as the narrator herself perceives:
I saw the offer as a sign of forgiveness, a tremendous burden removed.
That Jing-Mei sees this gift as the removal of a burden indicates that the weight of guilt and anger from her conflict with her mother has finally ended.
Jing Mei's discovery of the partner-song to "Pleading Child" indicates her own development as an individual and her arrival at a stage where she is happy with who she is and is no longer trying to be someone she is not or live her life for someone else (namely her mother). The title, "Perfectly Contented" clearly suggests that having gone through a stage where Jing Mei was a "Pleading Child", desperate for her mother's approval, she is now happy with herself.
Jing Mei's realisation that they were "two halves of the same song" perhaps indicates that this is a universal struggle that all must go through: we all go through a stage when we are a "Pleading Child", wanting our parents' approval and aprobation, yet eventually have to learn to live our own life and make our own choices, and become "Perfectly Contented."