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Choice D is correct. An equation defining a line in the xy-plane can be written in the form $y=mx+b$, where $m$ represents the slope and $\left(0,b\right)$ represents the y-intercept of the line. It’s given that line $k$ passes through the point $\left(0,-5\right)$; therefore, $b=-5$. The slope, $m$, of a line can be found using any two points on the line, $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, and the slope formula $m=\frac{y_2-y_1}{x_2-x_1}$. Substituting the points $\left(0,-5\right)$ and $\left(1,-1\right)$ for $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, respectively, in the slope formula yields $m=\frac{\left(-1-\left(-5\right)\right)}{\left(1-0\right)}$, or $m=4$. Substituting $4$ for $m$ and $-5$ for $b$ in the equation $y=mx+b$ yields $y=4x-5$. 

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

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

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