canada united-kingdom equity
A court exercising its equitable jurisdiction may disallow a minor from impeaching the validity of a contract on the grounds of minority if the minor acted fraudulently: Wilbur v. Jones (1881), 21 N.B.R. 4 (C.A.).
See also Stocks v. Wilson (1913), 2 K.B. 235:
What the Court of Equity has done in cases of this kind is to
prevent the infant from retaining the benefit of what he has
obtained by reason of his fraud. It has done no more than this,
and this is a very different thing from making him liable to pay
damages or compensation for the loss of the other party’s bargain.
If the infant has obtained property by fraud he can be compelled
to restore it; if he has obtained money he can be compelled to
refund it. If he has not obtained either, but has only purported
to bind himself by an obligation to transfer property or to pay
money, neither in a Court of law nor a Court of Equity can he
be compelled to make good his promise or to make satisfaction
for its breach.
Fraud in this context is not limited to literal misrepresentations at the time of contract formation: "support may be found for the view that the mere fact that the minor wishes to retain the property which he has obtained while at the same time pleading infancy as a defence to a claim for its value, is fradulent conduct in the requisite sense... 'for an infant to attempt to obtain something for nothing is, in effect, fraud in the eye of equity'": John D. McCamus, "Restitution of Benefits Conferred Under Minors' Contract" (1979) 28 U.N.B. Law Journal 89.
In the circumstances you describe, if all is believed by the court, the court may exercise its equitable jurisdiction to require the infant to return any benefit they obtained from the transaction.
I cannot find an example case, but it also seems open for the court to enjoin the minor to stop disclosing the secrets.
Common law and equity
user6726's answer focuses on the limited nature of remedies for this wrong at common law. user6726 is correct that fraudulent misrepresentation has a fairly narrow conception in common law, and it is a good answer for pointing this out. My answer is complementary in that it focuses on potential equitable remedies. This is yet another example where the distinction between common law and equity is significant: common law produces a somewhat harsh result for the wronged party, but equity may step in to return some fairness to the situation.
The common law rule is that a minor is not liable to restore benefits conferred on him under a contract which is unenforceable against him, even if the contract results from his fraudulent misrepresentation of majority.
Chitty on Contracts, §11-056.