Suppose that $M$ is a semisimple module. I want to show that $J(M)=0$
If $N$ is a simple submodule of $M$, then $M=N \oplus N'$ and $M/N'$ is isomorphic to $N$ and so is simple. Then $N'$ must be maximal, so it contains $J(M)$. Then $J(M)\cap N=(0)$.
Since $M$ is semisimple, so it is a direct sum of simple modules, each of which intersects $J(M)$ in (0).
Why $J(M)\cap M=(0)?$
You are almost there, since $M$ is semisimple we have
$$ M = \bigoplus_{i \in I}S_i $$
with each $S_i$ simple. Pick $x \in M \cap J(M)$ and write
$$ x = s_1 + \cdots + s_n $$
with $s_i \in S_{j_i}$ and $j_1, \dots, j_n \in I$. Since $S_{j_i}$ is simple for each $i$, we have that $\bigoplus_{I \setminus \{j_i\}}S_i$ is maximal and thus contains $J(M)$. However, since $x \in J(M)$, in particular this would show that
$$ s_i = x-(s_1+\dots+\widehat{s_i}+\dots+s_n) \in \bigoplus_{I \setminus \{j_i\}}S_i $$
which implies $s_i = 0$, and so $x = 0$.