In the paper http://arxiv.org/abs/1303.3328 by Samik Basu and Somnath Basu, it is claimed (Theorem A) that the homotopy groups of a simply connected closed 4-manifold $M$ are determined by the second Betti number $k$. In particular, if $k \geq 1$ and $j \geq 3$, $\pi_j (M) = \pi_j (\#^{k-1} S^2 \times S^3)$.
For example, using what we know about homotopy groups of spheres, they prove (Corollary 4.10) that if the second Betti number of $M$ is $k+1$ (sorry for the shift from $k$ to $k+1$, I keep the notations of the paper) then
$\pi_3(M) = \mathbb{Z}^{k(k+3)/2}$
$\pi_4(M)=\mathbb{Z}^{(k-1)(k+1)(k+3)/3} \oplus (\mathbb{Z}_2)^{2k}$
For a K3 surface $M$, $k+1=22$ so
$\pi_3(M) = \mathbb{Z}^{252}$
$\pi_4(M) =\mathbb{Z}^{3520} \oplus (\mathbb{Z}_2)^{42}$