To schedule an away at Lenox Hill, I would contact either Pat Kennemur in Dr. Ranawat's office (212) 434-2710, or Dr. Neil Roth (a sports/shoulder surgeon who is the coordinator for medical student education).
Rotators are not assigned to a particular service. You have the opportunity to work with many attendings, but you can also narrow your focus somewhat if you are looking for a letter. Dr. Hershman is the program director, so many students try to spend some quality time with him. Dr. Nicholas is another big name who enjoys having students and is good to work with. Dr. Roodriguez is one of our best educators, and students have really enjoyed working with him.
Housing in Manhattan can be a challenge. I recommend trying Craigslist, friends, etc. If you get desparate, you could ask Pat Kennemur to contact the residents. We get sent off on away rotations so there may be someone who would allow you to sublease their apartment for the month.
Good luck.