Is it possible to get (an estimation of) reaction rates from the Michaelis constant?
$K_M$ is a ratio and for QSSA it is $\dfrac{k_{cat}+k_r}{k_f}$ and for equilibrium approximation it is $\dfrac{k_r}{k_f}$.

It is clear from the formula that you cannot obtain all the constants simulataneously. If you know two (in case of QSSA) you can get the third.