Polylog dimensional subspaces of $\ell_\infty^N$

We show that a subspace of of $\ell_\infty^N$ of dimension $n>(\log N\log \log N)^2$ contains $2$-isomorphic copies of $\ell_\infty^k$ where $k$ tends to infinity with $n/(\log N\log \log N)^2$. More precisely, for every $\eta>0$, we show that any subspace of $\ell_\infty^N$ of dimension $n$ contains a subspace of dimension $m=c(\eta)\sqrt{n}/(\log N\log \log N)$ of distance at most $1+\eta$ from $\ell_\infty^m$.