1. Simplify the given expression

Given $f(a)=g(a)=k.$ The limit is $\lim_{x\to a}\frac{f(a)g(x)-f(a)-g(a)f(x)+f(a)}{g(x)-f(x)}=4.$

Now substitute $f(a)=g(a)=k$: $f(a)g(x)-f(a)-g(a)f(x)+f(a)=kg(x)-k-kf(x)+k=kg(x)-kf(x).$ So numerator becomes $k(g(x)-f(x)).$

Hence the expression reduces to $\lim_{x\to a}\frac{k(g(x)-f(x))}{g(x)-f(x)}.$ For $x\ne a$ near $a$, this is simply $\lim_{x\to a} k = k.$

Therefore, $k=4.$

2. Why the condition about nth derivatives is given

The condition that for some $n$, $f^{(n)}(a)$ and $g^{(n)}(a)$ exist and are not equal ensures that $g(x)-f(x)$ is not identically zero near $x=a$, so the denominator is meaningful in a deleted neighborhood and the cancellation is justified.

3. Compare with options

The value of $k$ is $4$.

So the correct option is: $\boxed{\text{B}}$

4. Comparison with stored answer

Stored correct answer: B

Our derived answer: B

They agree.