Choice A is correct. The value of $k$ for which $f\left(k\right)=3k$ can be found by substituting $k$ for $x$ and $3k$ for $f\left(x\right)$ in the given equation, $f\left(x\right)=\left|59-2x\right|$, which yields $3k=\left|59-2k\right|$. For this equation to be true, either $-3k=59-2k$ or $3k=59-2k$. Adding $2k$ to both sides of the equation $-3k=59-2k$ yields $-k=59$. Dividing both sides of this equation by $-1$ yields $k=-59$. To check whether $-59$ is the value of $k$, substituting $-59$ for $k$ in the equation $3k=\left|59-2k\right|$ yields $3\left(-59\right)=\left|59-2\left(-59\right)\right|$, which is equivalent to $-177=\left|177\right|$, or $-177=177$, which isn't a true statement. Therefore, $-59$ isn't the value of $k$. Adding $2k$ to both sides of the equation $3k=59-2k$ yields $5k=59$. Dividing both sides of this equation by $5$ yields $k=\frac{59}{5}$. To check whether $\frac{59}{5}$ is the value of $k$, substituting $\frac{59}{5}$ for $k$ in the equation $3k=\left|59-2k\right|$ yields $3\left(\frac{59}{5}\right)=\left|59-2\left(\frac{59}{5}\right)\right|$, which is equivalent to $\frac{177}{5}=\left|\frac{177}{5}\right|$, or $\frac{177}{5}=\frac{177}{5}$, which is a true statement. Therefore, the value of $k$ for which $f\left(k\right)=3k$ is $\frac{59}{5}$.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.