We have stopped assimilating them, but that does not mean that we could not if we put proper policies in place, including civics education for all immigrants, language requirements, and monolingual instruction and services.
We have made it too easy for them to not assimilate.
But the Muslims cannot be assimilate no matter what we do. In this, they are very different and very undesirable.
Many--probably most--immigrants from Latin America and East Asia and even sub-Saharan Africa actually want to assimilate into American culture. Legal or illegal, these immigrants--and especially their children and grandchildren--see this country as their future and embrace our founding ideals, our capitalist economy, our language, our Christian religion, our military struggle against the Islamofascist enemy, and our common future. In this era of liberalism, tolerance, and diversity, however, assimilation can prove difficult, particularly for immigrants who live in ethnic enclaves, especially in border towns where immigrants from a particular country comprise a majority of the population.
In some respects, ethnic enclaves differ little from those of the heyday of immigration in the late 1800s/early 1900s. Large Italian, Spanish, Greek, Polish, German, and other ethnic neighborhoods dotted New York City. These neighborhoods today have yielded to Dominican, Korean, Chinese, Mexican, Colombian, Ethiopian, and other new ethnic enclaves. They offer new immigrants a place where they can converse with others who share their former nationality, its culture, and its language. Learning English in a non-Anglophone foreign land can prove almost impossible; therefore, most immigrants begin understanding the language only after they arrive in America, and fluency can require years of practice.
Of course in the "more civilizationally confident era" of a century ago (as Mark Steyn would say), various societies attempted to enroll immigrants in programs designed to assimilate them into American society. These programs went beyond an occasional simple literacy class for those lucky enough to find an open seat at an acceptable time. They went so far as to include cooking classes that taught housewives the art of serving American foods according to American custom. Others taught immigrants in our ways of fashion, in imitating our peculiar accents, other proper housekeeping techniques, and various other aspects of American culture that we often overlook. And America was as divided and provincial a land then as it is now, between the agricultural homesteads of the prairie states to the Solid South to the industrial Midwest to the booming ports of the Eastern Seaboard to the Mormon Mountain states and the Pacific coast.
PS: "Civics instruction" should occur in the public schools, including economics, American history, world history, and quite possibly religion/theology (if it's still Constitutional to portray Christians as morally superior to crazy Muslims).