One way to approach this is to right down what you know. The maximum speed occurs when the displacement (call it x) is zero, and the maximum acceleration occurs when the displacement is at its maximum, the amplitude of the motion, call it X. You can right down an expression for the maximum acceleration
a = kX/m
at that point, the speed is zero and all its energy is potential. At its maximum speed, all the energy is kinetic. You can equate these two:
1/2mV^2 = 1/2kX^2
I think if you divide one equation by the other you'll get an expression for X in terms of things you know. From that you can find k and the angular frequency.
