  In 1992 Wirthm\"{u}ller showed that for any irreducible root system not of
type $E_8$ the ring of weak Jacobi forms invariant under Weyl group is a
polynomial algebra. However, it has recently been proved that for $E_8$ the
ring is not a polynomial algebra. Weyl invariant $E_8$ Jacobi forms have many
applications in string theory and it is an open problem to describe such forms.
The scaled refined free energies of $E$-strings with certain $\eta$-function
factors are conjectured to be Weyl invariant $E_8$ quasi holomorphic Jacobi
forms. It is further observed that the scaled refined free energies up to some
powers of $E_4$ can be written as polynomials in nine Sakai's $E_8$ Jacobi
forms and Eisenstein series $E_2$, $E_4$, $E_6$. Motivated by the physical
conjectures, we prove that for any Weyl invariant $E_8$ Jacobi form $\phi_t$ of
index $t$ the function $E_4^{[t/5]}\Delta^{[5t/6]}\phi_t$ can be expressed
uniquely as a polynomial in $E_4$, $E_6$ and Sakai's forms, where $[x]$ is the
integer part of $x$. This means that a Weyl invariant $E_8$ Jacobi form is
completely determined by a solution of some linear equations. By solving the
linear systems, we determine the generators of the free module of Weyl
invariant $E_8$ weak (resp. holomorphic) Jacobi forms of given index $t$ when
$t\leq 13$ (resp. $t\leq 11$).
