I have been tasked with proving that for all $\epsilon >0$ there exists some $C_\epsilon$ such that every finite metric space $X$ embeds into $\ell_{C_\epsilon\log(|X|+1)}$ with distortion at most $1+\epsilon$ where the distortion of an embedding $f$ is defined as \begin{align*} D(f) &= \sup\limits_{x\neq y} \frac{d_Y(f(x),f(y))}{d_X(x,y)}\cdot \sup\limits_{u\neq v} \frac{d_X(u,v)}{d_Y(f(u),f(v))} \end{align*} where in this case, $Y = \ell_{C_\epsilon\log(|X|+1)}$.
Here is what I have so far. First we let $M = \max d(x,y)$ and $m = \min\limits_{x\neq y} d(x,y)$ which are well-defined since $X$ is a finite metric space. For every embedding $f: X\to \ell_p$ we will have that \begin{align*} D(f)^p &= \frac{M^p}{m^p}\cdot \frac{\max{d_{\ell_p}(f(x),f(y))}}{\min\limits_{u\neq v}{d_{\ell_p}(f(u),f(v))}}\\ &= \frac{M^p}{m^p}\cdot \frac{\max\sum |f_i(x)-f_i(y)|^p}{\min\limits_{u\neq v}\sum |f_j(u)-f_j(v)|^p}\\ &\leq \frac{M^p}{m^p}\cdot \frac{|X|\max |f_i(x)-f_i(y)|^p}{\min\limits_{u\neq v}\sum |f_j(u)-f_j(v)|^p} \end{align*} So I just have to find an appropriate embedding $f:X\to\ell_p$ and some sufficiently large $C_\epsilon$ so that if $p = C_\epsilon(\log|X|+1)$ we get that the above quantity is bounded above by $(1+\epsilon)^p$. However, I am having trouble finding both the appropriate embedding and consequently the appropriate $C_\epsilon$. In particular, I am having trouble connecting how $p$ will depend on the $|X|$ and how to deal with the fact that $C_\epsilon$ should work for all finite metric spaces, especially since $M$ and $m$ will be defined differently for different metric spaces and so the quantity $\frac{M^p}{m^p}$ has no upper bound.
I attempted to use the natural embedding $x\to (d(x,x_1), (x,x_2), \ldots)$ for $x_i\in X$, but I got an upper bound on $D(f)^p$ as $|X|\left(\frac{M}{m}\right)^{2p}$ which grows faster than $(1+\epsilon)^p$. Any help would be appreciated.