The whole thing is dumb if you accept a premise of "infinite monkeys". An infinite number of monkeys will type the works of shakespeare immediately, because an infinite number of them will start with the very first key they hit and continue until the end. (So it'll be complete exactly as fast as a monkey can type it, typing as fast as simianly possible, with no mistakes.) You don't even need the infinite time.

It only becomes interesting if you look at the finite scenarios.

And BTW, the lifespan of the universe is finite due to the eventual decay of all matter, including the monkeys and the typewriters. There's no infinite time.