Select 2 out of the 13 kinds of cards first then select 2 from each kind.
$\large \binom{13}{2} \times \binom{4}{2} \times \binom{4}{2}$
now the 5th card can be chosen from remaining 11 kinds.
Total cards of remaining 11 kinds = $52-8=44$
choosing 1 card out of 44 = $\large \binom{44}{1}$
$\text{Probability} =\Large \frac{ \binom{13}{2} \times \binom{4}{2} \times \binom{4}{2} \times \binom{44}{1}}{\binom{52}{5}} $ $\approx 0.0475$