This would have been faster and more interesting.
For reference, this theorem was and is often used to counter evolutionism, but those that try to use it in this way fail to grasp that evolution is not random but selects the fittiest according to survival ability. By setting the survival rule to 'proximity to Shakespeare' you'd reach the end result relatively quickly.
Unless I'm missing something these seem to be two completely different experiments. One saying that a monkey given enough time could randomly create Shakespeare, and one saying given enough monkeys making random words they could eventually check off all words in a Shakespeare play.
The former problem you state is the correct one - that somewhere, one of those infinite monkeys, somewhere on his infinite typewriter paper, will contain a single consecutive string of the entire works of Shakespeare. Obviously that is monsterously more difficult, and almost certainly falls into one of those "If every atom in the universe was a computer, it'd still take x^n billion years" type combinatorial explosions.
It seems more like an expensive way to capture media attention.