Kuratowski's Maximal Principle and Tukey's Lemma Context: self-study.
Source work: Smullyan and Fitting: Set Theory and the Continuum Problem (rev. ed. 2010) chapter $4$: Superinduction, well ordering and choice: $\S 5$: Maximal principles.
We are given a form of Kuratowski's maximal principle, paraphrased for clarity:

Let $S$ be a set which is closed under chain unions. Then every element of $S$ is a subset of a maximal element of $S$ under the subset relation.

(We have proved previously in S&F that the above follows from Axiom of Choice.)
Then we are also given a form of Tukey's lemma:

Let $S$ be a non-empty set of finite character. Then every element of $S$ is a subset of a maximal element of $S$ under the subset relation.

In this context "of finite character" means:

Let $A$ be a class. $A$ is of finite character iff for all sets $x$, $x \in A$ iff every finite subset of $x$ is in $A$.

We are given this Lemma $5.4$:

Suppose $A$ is of finite character. Then: (1) $A$ contains, with each element $x$ all subsets of $x$ as well (that is, $x$ is swelled); (2) $A$ is closed under chain unions.

Now we are given Proposition $5.5$, where we have:

From this lemma we have the following: Kuratowski's maximal principle implies Tukey's lemma.

From the definition of Kuratowski's maximal principle and Tukey's lemma, it appears that all we need to do is invoke the result that a class which is closed under chain unions is of finite character.
But this is not what we have. Lemma $5.4$ says: "A class of finite character is closed under chain unions."
Hence this seems to me that this shows that Tukey's Lemma implies Kuratowski's Maximal Principle, not the other way round.
What have I missed?

I'm not sure what is confusing you here. The proof of Proposition 5.5 is immediate.
Assume Kuratowski. We prove Tukey. So let $S$ be a non-empty set of finite character. By Lemma 5.4, $S$ is closed under chain unions. By Kuratowski, every element of $S$ is a subset of a maximal element of $S$ under the subset relation. Done!
