I suppose the answer to this question relates to what you precisely mean by the word "adjust." If you mean does she become American and leave her Chinese part behind, then definitely the answer is no. However, if your question refers to the way in which immigrants out of necessity have to form some kind of compromise between their cultural heritage and the culture of their new country, then we can say that Jing-Mei's mother does adjust to America in this sense.
We need to remember that for any immigrant, life is very difficult as you have to adapt to a radically different set of cultural norms and values. You cannot carry on living your life as you did in your former country, and yet you are not able to completely leave your life behind either, as you are indelibly marked by your culture. Jing-Mei's mother does her best to live in America but also tries to bring up her daughter with Chinese values and culture. Consider how she responds to Jing-Mei's act of defiance in the story:
"Only two kinds of daughters," she shouted in Chinese. "Those who are obedient and those who follow their own mind! Only one kind of daughter can live in this house, Obedient daughter!"
Clearly, the fact that she is speaking in Chinese to her daughter and the way in which she demands obedience from Jing-Mei indicates that, whilst she is able to live in America, she is still very definitely Chinese and still looks at life from her Chinese perspective.