The lattice $L_{K3}=H^2(K3,\mathbb Z)$ is $2E_8+3U$, with $E_8$ negative definite and $U$ the hyperbolic lattice for the bilinear form $xy$. It is unimodular and has signature $(3,19)$.

The 16 (-2)-curves $E_i$ form a sublattice $16A_1$ of determinant $2^{16}$. It is not primitive in $L_{K3}$. The primitive lattice $K$ containing it is computed as follows. Consider a linear combination $F=\frac12\sum a_i E_i$ with $a_i=0,1$. Recall that $E_i$ are labeled by the 2-torsion points of the torus $A$, i.e. the elements of the group $A[2]$.

Then $F$ is in $K$ $\iff$ the function $a:A[2]\to \mathbb F_2$, 
$i\mapsto a_i$, is affine-linear. You will find the proof of this statement in Barth-(Hulek-)Peters-van de Ven "Compact complex surfaces", VIII.5.
(The element $\frac12\sum E_i$ in your example corresponds to the constant function 1, which is affine linear).
Thus, $K$ has index $2^5$ in $16A_1$ and its determinant is $2^{16}/(2^5)^2=2^6$.

$K$ is called the *Kummer lattice*. By the above, it is a concrete negative-definite lattice of rank 16 with determinant $2^6$. Nikulin proved that a K3 surface is a Kummer surface iff $Pic(X)$ contains $K$.

The orthogonal complement $K^{\perp}$ of $K$ in $L_{K3}$ is $H^2(A,\mathbb Z)$ but with the intersection form multiplied by 2. As a lattice, it is isomorphic to $3U(2)$. It has determinant $2^6$, the same as $K$. The lattice $L_{K3}=H^2(K3,\mathbb Z)$ is recovered from the primitive orthogonal summands $K$ and $K^{\perp}$.

However, your question has "Picard lattice" in the title. The Picard group of $X$ is strictly smaller than $H^2(X,\mathbb Z)$. To begin with, it has signature $(1,r-1)$, not $(3,19)$. For a Kummer surface, it contains Kummer lattice $K$ described above, and its intersection with $K^{\perp}$ is the image of the Picard group of $A$. For a Kummer surface one has $r=17,18,19$ or 20.

For the Mori-Kleiman cone of effective curves, which you would need for Gromov-Witten theory, the description you put in a box is already the best possible.