From electrical theory, the resonant frequency = 1/2π√LC
The mechanical equivalent is 1/2π√k/Meff
You need to first find the effective mass of the system. I don’t think your approach at that is right.
The effective mass can be quite complicated and you may need to use a Lagrangian.
I am not entirely sure of the correct method but if I have the time I may look into it as I enjoy these types of problems. You might have a look here for some inspiration, particularly the last part on the inverted pendulum with an oscillatory base.
Out of curiosity, is this an academic exercise or a practical exercise? In other words, does that design have some practical application as perhaps in seismology?