My first and foremost (albeit philosophical) advice would be to question yourself do you absolutely need an LLM since it is a significant investment of resources and time, so it is only logical that you question if you really need to do an LLM.
I would recommend studying and learning as much as possible, and networking with people, both in and out of the university in one’s filed, as two most important aspects of an LL.M. program.
International Commercial Arbitration was taught by four different professors, with different cultural and work backgrounds. Interacting with Professor Gerhard Wagner (my thesis supervisor) was in itself an experience.
Apart from UMich, I looked at Oxbridge, Harvard, Yale, Columbia, Chicago and Berkeley. Of these, I received acceptances from Berkeley and Michigan, while UChicago placed me on their waitlist.