To see that the set of ineffables is stationary, let $C\subseteq \eta_\omega$ be any closed and unbounded set. It suffices to show that the good indiscernibles arising in the definition of $\omega$-Erdos for the structure $(V_{\eta_\omega}, \in, C)$ are in $C$ since by what you have already shown such indiscernibles can be critical points of a non-trivial elementary embedding, and hence are ineffable.  However if $\gamma$ is such an indiscernible then $$(V_{\gamma}, \in, C\restriction \gamma)\prec (V_{\eta_\omega}, \in, C).$$

By elementarity, as $C$ is unbounded in $\eta_\omega$, then $C\restriction \gamma$ is unbounded in $\gamma$.  That is, $\gamma \in C$.

[ Suppose $\mathcal{A} = \langle L_\delta[A],\in,A,B\rangle $ is a
structure. $I\subseteq \delta$ is a *good sequence of indiscernibles*
for $\mathcal{A}$ if for all $\gamma\in I$:

(i) $\langle L_\gamma[A\cap \gamma],\in,A\cap \gamma,B\cap \gamma\rangle \prec \mathcal{A}$;

(ii) $I\backslash \gamma$ is a set of indiscernibles for the structure
$\langle L_\delta[A],\in,A,B, \langle \xi\rangle_{\xi \leq \gamma}\rangle$.

  We say that $\delta$ is $\tau$-*Erdos*  if every first order structure $\mathcal{A}=\langle L_\delta[A],\in,A,B\rangle $ has a good sequence of indiscernibles of o.t. $\tau$.  It can be shown by a combinatorial argument that each of the $\eta_\alpha$ as defined by Jech are $\alpha$-Erdos.

An equivalent definition of being $\tau$-Erdos is the following: let $C\subseteq \delta$ be c.u.b. Let $F:[C]^{<\omega}\rightarrow \delta$ be any *regressive function* (*i.e.* $F(a)<min(a)$ for all $a\in [C]^{<\omega}$. Then there is a set $I\subseteq\delta$ of o.t.($\tau$) which is homogeneous for $F$: $\forall n \, |F$ " $[I]^n| = 1$. 

The notion of good indiscernibility was introduced by Jensen in:

H.-D. Donder, R.B. Jensen, B. Koppelberg, Some applications of K , in: R. Jensen, A. Prestel (Eds.), Set Theory and Model Theory, in: Springer Lecture
Notes in Mathematics, vol. 872, Springer-Verlag, 1981, pp. 55–97. ]