Irregularity of $\{ w_1 aa w_2 \mid |w_1| \neq |w_2| \}$
For $i \ge 0$ define $w_i = b^i aa$.
For any $i,j \ge 0$ with $i \neq j$ you have that $b^i$ is a distinguishing extension for $w_i$ and $w_j$. Indeed, $w_i b^i \not\in L_2$ but $w_jb^i \in L_2$.
Then the number of equivalence classes of the set $\{ w_i \mid i \ge 0\}$ with respect to the equivalence relation &quot;having a distinguishing extension&quot; is not finite and, by the Myhill-Nerode theorem, $L_2$ is not regular.

Here is an alternative solution that uses closure properties and the pumping lemma following the hint of Hendrik Jan.
Suppose that $L_2$ is regular. Then so is $L' = \overline{L}_2 \cap \{b^i aa b^j\} = \{b^i aa b^i\}$, which is easily shown to be non-regular using the pumping lemma.
Indeed, suppose that $L'$ was regular, and let $p$ be its pumping length. For some $k \in \{1, \dots, p\}$, the word $b^p aa b^p \in L'$ can be written as $b^k b^{p-k} aa b^p$ such that $b^{ki} b^{p-k} aa b^p \in L'$ for every $i \ge 0$. Nevertheless, for $i=0$ we have $b^{p-k} aa b^p \not\in L'$, yielding a contradiction.