How does the piano symbolize the conflict between Jing Mei and her mother as well as the conflict within the mother herself?

You have asked two questions there - rules are you can only ask one, so I will respond to the first, which will include some heavy hints about the second... :-)

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.

From the first, it is clear that playing the piano is another one of Jing-Mei's mother's schemes to force her daughter into becoming a child prodigy. When Jing-Mei is told about her classes, she "felt as though I had been sent to hell." Her response to her mother clearly displays how she views what is happening: "Why don't you like me the way I am?" She sees her mother's schemes as a reflection on herself and feels that she is not accepted by her mother if she cannot be a genius.

It is only when she hears her mother bragging that Jing-Mei decides to "stop her foolish pride" with the dramatic climax of the story in the concert, beginning a catalogue of choices or "failures" where Jing-Mei asserted "my own will, my right to fall short of expectations."

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.