Given the functions $$ f(x)=\frac{x}{\ln x}, $$ and $g(x)=k(x-1)$ where $k \in \mathbf{R}$.
(1) Prove that for any $k \in \mathbf{R}$, the line $y=g(x)$ is not a tangent to the curve $y=f(x)$;
(2) If there exists $x \in[\mathrm{e}, \mathrm{e}^{2}]$, such that $$ f(x) \leqslant g(x)+\frac{1}{2}, $$ find the range of values for $k$. Please translate the following text into English.
And give the value of \(2k\) when \(k\) takes the minimum value. 

Let's think step by step and output the final answer within \boxed{}.
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