Obviously, everyone will differ in their interpretation of this story and what is most important, but to me you will want to focus on the ending of this challenging short story about growing up and gaining independence from your parents.
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.
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." The use of the names of these two pieces of music therefore to me encapsulate the message of this powerful work of fiction.