It is hard for people to hear things that are not significant in their native languages. I have had such an experience when attempting to learn a little Mandarin (very hard to hear tones), and seen it in other learners; for example native speakers of Spanish often do not hear the glottal stops that occur in AmE, because the glottal stop does not exist in Spanish.
Your problem is compounded by the fact that your student has been in America so long. I think the only solution is one-on-one teaching. You have to go right back to the alphabet and the sound system, and insist that every letter be pronounced.
I believe that in some dialects of South American Spanish syllables are often swallowed in a similar way, and we have some teachers here who are native speakers of Spanish. Maybe they can help.
Student or Learner