Perfect closure is perfect

The answer to your question is, that it isn't necessarily true that $K_F^p$ is a perfect field. For example, let $F$ be any non-perfect field (e.g. $\mathbb{F}_p(T)$), and let $K=F$, so that we must have $K_F^p=F$.

When $K=\bar{F}$, an algebraic closure of $F$, then $K_F^p$ is the smallest perfect subfield of $K$ containing $F$, hence the name "perfect closure". This is Lemma 3.16 in Karpilovsky's Field Theory (for some reason I couldn't find this in a more standard reference like Lang). As you mention, $K_F^p$ consists of those $\alpha\in K$ such that $\alpha^{p^n}\in F$ for some $n$, and note that the field that Karpilovsky refers to as the perfect closure is $$F^{p^{-\infty}}=\{a\in\bar{F}\mid a^{p^n}\in F \text{ for some }n\geq0\}.$$