This is a good point. At best you can figure out that period is proportional to (not equal to) sqrt(M/K) multiplied by some function of other parameters, say, one involving displacement and another characterizing the non-linearity (if K is just the initial slope, as I've seen done before). It's a fortunate coincidence if the other parameters are unimportant. You can not determine based solely on dimensional analysis whether certain parameters are unimportant.