I think Verbiski proves a refined form of the Mirror Conjecture for hyperkaehler manifolds, not the conjecture in the strict form. This is explained at page 3 of the paper that you link.

In fact, the assumptions for the standard conjecture are known to hold only for a hyperkaehler manifold which is generic in its deformation class. More precisely, Verbitski proves the following result, see page 26.

Theorem 5.4. Let $M$ be a compact holomorphically symplectic manifold, which is generic in its deformation class. Then the Mirror Conjecture holds for $M$, which is mirror dual to itself.

For instance, $\mathrm{Pic}(M)=0$ is enough, see the footnote in the same page. I guess that the examples of K3 surfaces with non-trivial mirror do not satisfy this condition.

I think Verbisky proves a refined form of the Mirror Conjecture for hyperkaehler manifolds, not the conjecture in the strict form. This is explained at page 3 of the paper that you link.

In fact, the assumptions for the conjecture are known to hold only for a hyperkaehler manifold which is generic in its deformation class. More precisely, Verbitsky proves the following result, see page 26.

Theorem 5.4. Let $M$ be a compact holomorphically symplectic manifold, which is generic in its deformation class. Then the Mirror Conjecture holds for $M$, which is mirror dual to itself.

For instance, $\mathrm{Pic}(M)=0$ is enough, see the footnote in the same page.