The examples you give:

import mkl_random
rs = mkl_random.RandomState(seed=777, brng="MT19937")

The following error pops up after I run it:
AttributeError: module 'mkl_random' has no attribute 'RandomState'

I changed one of the import methods after I tried it and it worked:

from mkl_random import mklrand
rs = mklrand.RandomState(seed=777, brng="MT19937")
random_integers = rs.randint(0, 10, size=10)

I don't know if that's right. Maybe you should update the documentation?