What would have been the difficulties for the US in such a move? What would have been the easiest part for them?
Was there a strong push for annexation? Who wanted such an annexation?
There was an All of Mexico Movement that advocated annexing all of Mexico however it was not very strong and dissipated at the end of the Mexican-American War. The large size of Mexican states would have made them more powerful than some existing American ones and it faced political opposition for a number of reasons including voting rights and racism. It's is covered more here https://books.google.com/books?id=DSzyMGh8pNwC&pg=PA45&lpg=PA44&ots=6DUkFcc4NX&focus=viewport&dq=all+of+mexico+movement&output=html_text