First dispose off the cases $n=2$ and $n=3$ by explicit verification. We will henceforth assume $n \geq 4$.
Since the function $$ f(x) = \frac{x}{\ln x} $$ is increasing in $[e, \infty)$, it follows that $$ \frac{k}{\ln k} \leq \frac{n}{\ln n} \tag{1} $$ for $3 \leq k \leq n$. Also, assuming $n \geq 4$, we have $$ \frac{2}{\ln 2} = \frac{4}{\ln 4} \leq \frac{n}{\ln n}. \tag{2} $$ Adding the inequalities $(1)$ (for $3 \leq k \leq n$) and $(2)$, we get $$ \sum_{k=2}^{n} \frac{k}{\ln k} \leq \frac{n(n-1)}{\ln n}, $$ which implies the result.