For simplicity, it is common to work just with the language of equality. Let $V^2$ be the set of second-order validities in this language. It is also somewhat common to look at the set $V^2$ instead of the set of consistent sentences. Of course a sentence is consistent if and only if its negation is not a validity, so there is no real difference in studying $V^2$.
The answer by Henning Makholm shows that $V^2$ is not definable in first-order arithmetic.
This can be extended to show that $V^2$ is not definable in second-order arithmetic. The proof is essentially just Tarski's theorem on the undefinability of truth.
Because each $n$th-level higher order arithmetic is interpretable in second-order arithmetic in a well-known way, this shows that $V^2$ is not definable in $n$th order arithmetic for any $n$.
I don't have a copy at hand of "Set Theory and Higher-Order Logic", Richard Montague, 1965, in Formal Systems and Recursive Functions, Studies in Logic and the Foundations of Mathematics v. 40, pp. 131-148. DOI 10.1016/S0049-237X(08)71686-0. Shapiro attributes to this paper an extension that $V^2$ is not definable even in a collection of transfinite levels of higher-order arithmetic.
As an upper bound, it appears Montague proved that if $\lambda$ is the Lowenheim number of second-order logic then $V^2$ is definable in $(\lambda + 1)$th order arithmetic. The Lowenheim number is the smallest cardinal $\lambda$ so that if a theory $T$ has a model then it has a model of size less than $\max(|T|,\lambda)$.
For second order logic in standard semantics the Lowenheim number is known to be extremely large - it is larger than the first measurable cardinal if there is a measurable cardinal. For more discussion of $\lambda$, which they call $\text{LS}(L^2)$, see Menachem Magidor and Jouko Väänänen, "On Löwenheim-Skolem-Tarski numbers for extensions of first order logic", J. Math. Log. v. 11, 2011,
DOI 10.1142/S0219061311001018,
