The thought experiment is underspecified in ways that undermine it.
How big is the book?
Seems like a nitpick about a theoretical concession in a thought experiment, right? No, it's actually very important.
If the book contains simple instructions like if you receive character A then reply with B, etc. in a Choose-Your-Own-Adventure style, then it would have to be exponentially large. Too large to be able to carry out more than a few words of conversation, and certainly you would not be able to converse about math, e.g, you're asked what 一 plus 二 is and you say 三, etc. The book would rapidly become larger than the planet Earth with any reasonable conversation depth. This is not just practical problem.
Okay, so the book isn't a simple lookup table, fine. You'll have to have a piece of scrap paper and write down things to refer to them later. But once you do that, it's obvious that you've created a system of memory. It undermines the whole force of the thought experiment. The book was supposed to contain all the brains and you were just supposed to mechanically follow along without understanding what you're doing. But now you're doing complicated things like solving math problems and then converting the answers into Chinese. In order to give the book a finite size, we've given you a lot of work to do, and now it's totally reasonable to say that you do know how to write Chinese. You write Chinese by looking it up, the same way that real translators do!
The Chinese room thought experiment is much discussed but it's a pretty poor thought experiment. It handwaves away all the important parts of language in order to make an inscrutable point about machine intelligence. It neither sheds light on machine intelligence nor language.