For the audience, $E(1)$ is the underlying smooth 4-manifold for a rational elliptic surface. i.e. $\mathbb{C}P^2$ blown up 9 times, while $E(2)$ is the underlying smooth manifold for a $K3$ surface and is diffeomorphic to the symplectic fiber sum of two copies of $E(1)$ along an elliptic fiber. ($E(n)$ is the fiber sum of $n$-copies of $E(1)$)

I assume that you don't find the change of basis in $H_2(E(n))$ method in Gompf and Stipsciz sufficient for your purposes? An alternative way of finding the intersection forms is to look at Kodira's classification of singular fibers in an elliptic fibration. (See for example Barth-Hulek-Peters-Van de ven.) From this we can obtain an elliptic fibration with two or three singular fibers one of which has intersection form $-E8$, the others being either a cusp or two nodes. The regular neighborhood $W$ of the $-E8$ singular fiber is diffeomorphic to a plumbing of $-2$-spheres in a configuration given by the Dynkin diagram for $E8$ and has boundary equal to the Poincare homology sphere $\Sigma(2,3,5)$. The intersection for the other side, called the nucleus $N(1)$, is $\left(\begin{matrix} 0 & 1 \\ 1 & -1 \end{matrix}\right)$.

The homology of $E(1)$ is generated by $h,e_1,\ldots, e_9$ (the generators of $\mathbb{C}P^2$ and the $\overline{\mathbb{C}P}^2$s. Then $H_2(N(1))$ is generated by $[F]=3h-\sum_{i=1}^8 e_i$ and $e_9$ where $F$ is the elliptic fiber and $H_2(W)$ is generated by $e_1-e_2,e_2-e_3,\ldots,e_7-e_8,-h+e_6+e_7+e_8$.

You can now reconstruct your statement using the Meyer-Vietoris and relative homology sequences. The three square $0$ classes end up being the 3 "rim tori" in the $T^3$ boundary of $E(n)\setminus N(F)$.