Given data as in the statement we constructed a $4$–manifold $M_{\epsilon}$ and a $1$–parameter family of closed definite triples $\bm{\underline{\omega}}_{\epsilon}$ which are approximately hyperkähler. For $\epsilon$ sufficiently small we can apply Lemma 6.13 to find unique $\bm{\underline{a}}_{\epsilon}\in C^{1,\alpha}_{\delta}(T^{\ast}M_{\epsilon})$ for $\delta\in(-\tfrac{1}{2},0)$ and $\bm{\underline{\zeta}}_{\epsilon}\in\mathcal{H}^{+}_{\epsilon}$ such that $\|\bm{\underline{a}}_{\epsilon}\|_{C^{1,\alpha}_{\delta}}+\|\bm{\underline{\zeta}}_{\epsilon}\|\leq C\epsilon^{\frac{11-2\delta}{5}}$ and $\bm{\underline{\omega}}_{\epsilon}+d\bm{\underline{a}}_{\epsilon}+\bm{\underline{\zeta}}_{\epsilon}$ is a hyperkähler structure on $M_{\epsilon}$. In particular, since $b_{1}(M_{\epsilon})=0$ by Proposition 5.1, $M_{\epsilon}$ must be diffeomorphic to the K3 surface.

Away from the gluing regions $\bm{\underline{a}}_{\epsilon}$ solves the elliptic PDE $d^{+}\bm{\underline{a}}_{\epsilon}=\mathcal{F}(d^{-}\bm{\underline{a}}_{\epsilon}\ast d^{-}\bm{\underline{a}}_{\epsilon})-\bm{\underline{\zeta}}_{\epsilon}$, $d^{\ast}\bm{\underline{a}}_{\epsilon}=0$. By elliptic regularity, for any $k\geq 2$ the $C^{k,\alpha}$–norm of $\bm{\underline{a}}_{\epsilon}$ on compact sets of $M^{\textup{gh}}_{\epsilon}$ and (after rescaling) on compact sets of the gravitational instantons $M_{j}$ and $N_{i}$ is controlled in terms of $\|\bm{\underline{a}}_{\epsilon}\|_{C^{1,\alpha}_{\delta}}+\|\bm{\underline{\zeta}}_{\epsilon}\|$. In particular, on compact sets of $M^{\textup{gh}}_{\epsilon}$ the hyperkähler metric induced by $\bm{\underline{\omega}}_{\epsilon}+d\bm{\underline{a}}_{\epsilon}+\bm{\underline{\zeta}}_{\epsilon}$ is $C^{k,\alpha}$–close to $g_{\mathbb{T}}+\epsilon^{2}\theta^{2}$. The statements (i), (ii) and (iii) about the limit $\epsilon\rightarrow 0$ now follow. ∎