Step 1: Understand the condition for continuity

A function $f(x)$ is continuous at $x = a$ if the limit of the function as $x$ approaches $a$ exists and is equal to the value of the function at $a$. Mathematically, $\lim_{x \to -1} f(x) = f(-1)$.

Step 2: Evaluate the limit

We need to find $\lim_{x \to -1} \frac{x^{2}-4x-5}{x+1}$. Substituting $x = -1$ directly gives the indeterminate form $\frac{0}{0}$. We factorize the numerator:

Now, substitute this back into the limit:

Step 3: Solve for k

Evaluating the limit as $x \to -1$: $$-1 - 5 = -6$$ Since the function is continuous at $x = -1$, we must have $f(-1) = \lim_{x \to -1} f(x)$. Given $f(-1) = k$, we conclude $k = -6$.