Suppose $R$ is semisimple and all left simple ideals of $R$ are isomorphic.
I want to show that except $0$ and $R$, there is no two sided ideal in $R$.
Let $J$ be a two sided ideal. So it is a left ideal of $R$, hence it is a left $R$-module semisimple. So $J$ is a sum of simple left ideals of $J$.
My goal is to show that all simple left ideals of $J$, are also simple left ideals of $R$.
Let $I$ be the unique, up to isomorphism, simple left ideal of $R$. Let $D$ be the division ring $\operatorname{End}_R(I)$. By semi-simplicity of $R$ there is a positive integer $n$ with ${}_RR\simeq\bigoplus_{i=1}^nI$ and thus $R^\mathrm{op}\simeq\operatorname{End}({}_RR)\simeq M_n(D)$ i.e. $R\simeq M_n(D^\mathrm{op})$. Since for any ring $S$ one has a one-to-one correspondence between $\{$two-sided ideals of $S\}$ and $\{$two-sided ideals of $M_n(S)\}$ via $J\longrightarrow M_n(J)$, $R$, which is isomorphic to $M_n(D^\mathrm{op})$, has only two two-sided ideals: $0$ and itself.