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Hell, an actually infinite amount of monkeys would produce the complete works of Shakespeare plus some originals in the same style in the exact amount of time it took to literally press the necessary buttons.
An infinite amount of monkeys each given an infinite amount of time would produce all infinite strings possible on a typewriter (this includes ones that just happen to be terminated with a neverending substring of blank spaces, i.e. one where the monkey stops or presses whitespace keys and nothing else an arbitrary number of times).
If a cure for cancer exists and is expressable through language, they would not only produce one, but it would be there in every single language transliteratable to a Latin script; it would be there in ASCII art; it would be there in literally every text-based form imaginable. Of course the trouble would be sorting the infinite wheat from the infinite chaff.
Edit: https://libraryofbabel.info/ for the finite strings of 3200 characters.
I mean, even if the typewriter had an extra button setting off a bomb, killing the monkey and all the monkeys around them, they'd still pull it off. That's infinity baby.
The Library of Babel Jorge Luis Borges (1941)
A good read.
That is wrong assumption. We know that even when something is infinite it may never reach required value. Shakespeare or anything else may be a unique event in the infinite space-time universe.