A quantitative version of the commutator theorem for zero trace matrices

Let $A$ be a $m\times m$ complex matrix with zero trace and let $\text{e}>0$. Then there are $m\times m$ matrices $B$ and $C$ such that $A=[B,C]$ and $\mathrm{\parallel }B\mathrm{\parallel }\mathrm{\parallel }C\mathrm{\parallel }\le K_{\text{e}}{m}^{\text{e}}\mathrm{\parallel }A\mathrm{\parallel }$ where $K_{\text{e}}$ depends only on $\text{e}$. Moreover, the matrix $B$ can be taken to be normal.