Solution:

To see if any pattern exists, we can test out small values of $k.$

We have that $$8 \cdot 8 = 64,$$ $$8 \cdot 88 = 704,$$ $$8 \cdot 888 = 7104,$$ $$8 \cdot 8888 = 71104.$$

From this, it is pretty safe to guess that for every increment of $k,$ there is an extra $1$ added to the product. If you know how to use induction, you can prove that this pattern holds, but that's not necessary to solve the problem.

This means that for any $k \geq 3,$ the sum of the digits in the product is $$7 + 4 + 0 + k - 2 = k + 9.$$

Finally, we get $$k + 9 = 1000$$$$k = 991.$$

Thus, D is the correct answer.