January is traditionally a tough month for Broadway, and many shows close as early as January 2 when the new year arrives.
If BRIEF ENCOUNTER is open when you arrive, I strongly recommend it if you didn't catch it in the UK.
Also check out PLAYS! There are many non-musical plays which are quite good. Go see ANGELS IN AMERICA at Signature Theater Company. That will take up 7 hours of your free time which will end up being the highlight of your theatre-attending years thus far. ANGELS OF AMERICA is literally the best decision of a show that you could possibly make while here in the States.
If THE MERCHANT OF VENICE is still running, catch that, too.
"The Spectacle has, indeed, an emotional attraction of its own, but, of all the parts, it is the least artistic, and connected least with the art of poetry. For the power of Tragedy, we may be sure, is felt even apart from representation and actors. Besides, the production of spectacular effects depends more on the art of the stage machinist than on that of the poet."
--Aristotle