The correct answer is $\boxed{\frac{1}{{221}}}$.
The probability of event A happening, given that event B has already happened, is called the conditional probability of A given B, and is denoted by $P(A|B)$. In this case, event A is “the second card drawn is a king” and event B is “the first card drawn is a king”.
We can calculate the probability of both cards being kings by multiplying the probability of the first card being a king by the probability of the second card being a king, given that the first card is a king.
The probability of the first card being a king is $\frac{4}{{52}}$, since there are 4 kings in a standard deck of 52 cards.
The probability of the second card being a king, given that the first card is a king, is $\frac{3}{{51}}$, since there are 3 kings left in the deck after the first card is drawn.
Therefore, the probability of both cards being kings is $\frac{4}{{52}} \times \frac{3}{{51}} = \boxed{\frac{1}{{221}}}$.
Option A is incorrect because it is the probability of the first card being a king.
Option B is incorrect because it is the probability of the second card being a king.
Option C is incorrect because it is the probability of both cards being kings, if the first card is replaced after it is drawn.