How do "Pleading Child" and "Perfectly Contented" have a double meaning in the story "Two Kinds"? What does the last sentence of the story mean?

The two songs and the last line of the story illustrate the same idea. "And after I had played them both a few times, I realized they were two halves of the same song." The two songs symbolize the two parts of Jing-mei's life and psyche. The title, "Two Kinds," also plays into this notion of the double. 

First of all, Jing-mei is a Chinese-American girl. Her mother has certainly embraced the American spirit, notably in pushing her daughter to pursue the American Dream. But her mother's Chinese culture, the parental pressuring, and the domineering nature that may come from that culture are present to Jing-mei as a child. Jing-mei must deal with this while trying to be more individually free. And this idea of individual freedom was more celebrated in America at this time. So, apropos of the "two kinds" theme, Jing-mei must negotiate two cultures as she matures. 

But in particular reference to the two songs, the "Pleading Child" represents the time Jing-mei was tested and forced to practice piano as a child. When Jing-mei is finally free from this discipline, she becomes more individually free and feels more like the "Perfectly Contented" child/woman. In the last line, Jing-mei realizes that these two songs/identities are both a part of her, and with that realization, she feels whole. Likewise, she is both Chinese and American. Although, by adulthood, she is probably more American in many ways, she still has elements of both cultures. She has this sense that her unified identity is a series of doubles: Chinese/American, Pleading/Contented, and Traditional/Individual.