If you draw 26 cards from 52 cards, what is the probability that you get 4 kings?

You are correct. Your teaching assistant's error is in assuming that the events that each king goes to the first player are independent. They are not.

As Angina Sing noted, your TA is assuming that the $4$ events are independent, but they are not. If we correct his calculation, we arrive at the same answer you did. The probability that the first player is dealt the King of Spades is $\frac{26}{52}$. Once he has the King of Spades, what is the probability that he also receives the King of Hearts? There are $25$ spots remaining in his hand, and $51$ spots overall, so the probability that he is dealt both Aces is $\frac{26}{52}\cdot\frac{25}{51}$. (If you have learned about conditional probability, this is simply the fact that $\Pr(S\cap H)=\Pr(S)\Pr(H|S)$.) Continuing in this manner, we get that the probability that he gets all $4$ Aces is $$\frac{26\cdot25\cdot24\cdot23}{52\cdot51\cdot50\cdot49}$$ the same answer you got.