(You might want this to get double-checked by someone else; my probability is a bit rusty.)
Recall the formula for conditional probability:
$$ \begin{align}
P(A \cap B) &= P(A|B)P(B) \\
P(A|B) &= \frac{P(A \cap B)}{P(B)}
\end{align}$$
Here, let $A$ be the event that the $k$th card is the largest of the $m$ cards, and $B$ be the
event that the $k$th card is the largest of the $k$ cards drawn.
Since $A$ implies $B$, $P(A \cap B) = P(A)$. So the probability is $1/m$.
The probability of event $B$, that the $k$th card is the largest of $k$ cards, without knowing any other information, is $1/k$.
Then $$P(A|B) = \frac{1/m}{1/k} = \frac{k}{m}.$$