A proportional Dvoretzky-Rogers factorization result

If $X$ is an $n$-dimensional normed space and $\varepsilon\in(0,1)$, there exists $m\geq(1-\varepsilon)n$, such that the formal identity $i_{2,\infty}\colon l^m_2\to l^m_\infty$ can be written as $i_{2,\infty}=\alpha\circ\beta,\beta\colon l^m_2\to X,\alpha\colon X\to l^m_\infty$, with $\|\alpha\|\cdot\|\beta\|\leq c/\varepsilon$. This is proved as a consequence of a Sauer-Shelah type theorem for ellipsoids.