A space in which all compact subsets are closed is called KC-space.
A space in which every infinite set contains an infinite subset with only a finite number of accumulation points is said to have the finite derived set property.
Theorem: A hereditarily Lindelöf minimal KC-space is sequential.
Proof: Suppose that $(X, \tau)$ is a hereditarily Lindelof minimal $KC$-space; suppose that $A \subset X $ is not closed and hence not compact. Since $X$ is hereditarily Lindelof, $A$ is not countably compact and hence we can find a countable discrete subset $D = \{x_n : n ∈ \omega \} \subset A$ which is closed in $A$; that is to say, all of the accumulation points of $D$ lie outside of $A$. $X$ has the FDS-property, and so there is some countably infinite set $E \subset D$ with only a finite number of accumulation points in $X$, all of which lie in $\overline{A} - A$. Thus $\overline{E}$ is a countable, compact $KC$-space and $\overline{E}$ is sequential; thus there is a sequence in E converging out of $E$ and hence out of $A$.
Why is $\overline{E}$ is a countable, compact $KC$-space?
By Lemma 2.1 of
(which appears to be the paper you are reading) we have that every hereditarily Lindelöf, minimal KC-space is compact. Therefore $X$ is compact, and thus so, too, is any closed subspace of $X$, such as $\overline{E}$.
(As an aside, note that in
it was subsequently shown that all minimal KC spaces are compact.)
It is fairly easy to see that subspaces of KC-spaces are themselves KC.