Expected top rank among four dealt cards

Four cards are dealt from a well-shuffled standard deck. Counting aces as $1$, number cards at face value, jacks as $11$, queens as $12$ and kings as $13$, what is the expected value of the highest of the four cards, to two decimal places?

$\approx 10.95$ (exactly $\tfrac{32577}{2975}$). The maximum is at most $k$ exactly when all four cards come from the $4k$ cards of rank $\le k$, so $\mathbb{P}(\max\le k)=\binom{4k}{4}/\binom{52}{4}$. Then $\mathbb{E}[\max]=\sum_{k=0}^{12}\left(1-\mathbb{P}(\max\le k)\right)$.