In practice native speakers probably wouldn't use the word knew that way in such a context.
It's fine to say something like "I knew him back in the 80s when we were at college together", because the implication is you "knew" him for years (you might or might not still know him).
But for a period as short as 15 minutes, it really doesn't make a lot of sense. Firstly because that's barely enough time to "get to know" someone. Secondly, if we allow that you could do so, and if a few minutes later someone asked "Do you know Mr. Brown?" you'd presumably answer "Yes". Because you still know him, even if he's just left (unless, as John Lawler suggests, he just died :).
OP's "I had known Mr. Brown for exactly 15 minutes." is just about credible, but it seems a bit "dated/formal/literary" to me. More natural phrasing would be something like...
"I had only known Mr. Brown for 15 minutes."
Personally, I'd prefer met rather than known, for the reason given above. But that just creates more problems when OP uses the same verb with a slightly different sense in the next sentence.