This is a great question! In this short story we have observed the conflict between Jing-Mei and her mother from its highs to its lows. This of course finds its climax in the piano recital and the bitter argument that happens afterwards. At the end of the story we advance forward a few years to Jing-Mei as an adult, after her, as she puts it, "failing her mother so many times", but each time "asserting my own will, my right to fall short of expectations." It is when her mother gives her the piano that Jing-Mei begins to change in her attitude. She describes the piano using an interesting metaphor, "a shiny trophy", because she had won it on her own terms and not her mother's.
At the end, Jing-Mei receives the piano and she beings to play "Pleading Child" again. The last paragraph is worthy of some serious analysis:
And for the first time, or so it seemed, I noticed the piece on the right-hand side. It was called "Perfectly Contented." I tried to play this one as well. It had a lighter melody but the same flowing rhythm and turned out to be quite easy. "Pleading Child" was shorter but slower; "Perfectly Contented" was longer but faster. And after I played them both a few times, I realised they were two halves of the same song.
Jing-Mei realises that just as these two pieces of music go together inseparably, being "two halves of the same song", so in her life, the stage of "Pleading Child", which interestingly is described as short but slow, is inextricably linked to "Perfectly Contented", which was longer and faster. Jing-Mei, through her childhood was the "Pleading Child", wanting her mother's attention and praise, and now, as an adult, she has reached the stage of being "Perfectly Contented", knowing who she is as an adult and being happy in her identity. However, what she realises is that she can't have one without the other - both are irreplaceable parts of life's journey.