Let me assume that the theory of $L$ is an element of $L$. This
happens, for example, if $L_\alpha\prec L$ for some ordinal
$\alpha$, because in this case the theory of $L$ is the same as the
theory of $L_\alpha$, which is an element of $L$. Let $t$ be the theory of $L$,  which is (coded by) a real in $L$. This real is therefore the
$\alpha^{th}$ real in the $L$-order, and in order to define the
theory $t$, it will suffice to define the ordinal $\alpha$.

Let $L[G]$ be a forcing extension of $L$ forcing to collapse
$\aleph_{\alpha}^L$ to $\omega$. So in $L[G]$, the true $\omega_1$
is the same as $\omega_{\alpha+1}^L$, and we can determine this
inside $H_{\omega_1}$. In that structure, we can define the class
of ordinals that are cardinals in $L$, and there will be exactly
$\alpha$ of them.

This makes the theory definable by a formula quantifying only over
reals, and it will be $\Delta^1_n$ for some smallish $n$.

I'll think about the exact value of $n$ and post an update, unless
someone else wants to help me figure out what $n$ this gives.