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If I plan to apply to law schools next cycle (after taking the LSAT on September 2017) to attend on September 2018, what would be the exact deadline to get the LORs? I ask because, after asking my professor for a letter of recommendation, my professor asked me for the deadlines, and I'm not sure what to tell him.

I'm assuming the LOR deadline = Application deadline. So when exactly (month in particular) would that be ?

I would decide when you want to get your applications in by and then give your recommenders a deadline of a few weeks before that just in case they don't get it in by the deadline you give.