Given the spectrogram of an unknown signal embedded in a Gaussian noise, the Minimal Statistics Maximum Likelihood (MiniSMaL) estimator of the noise time-varying power spectrum is presented, and a method to tune one of its parameter is studied. The objective of the minimal statistics approach is to separate the signal of interest from the noise in order to estimate properly the probabilistic properties of the latter. Considering an initial time-frequency estimation neighborhood, the strategy relies on the selection of a minimal subset containing the time-frequency coefficients with the smallest values. Estimators of the noise are then sought from this minimal subset. In this work, the case of a spectrogram constructed from a finite-length discrete-time noisy signal is presented. This study extends previous works on minimal statistics on two aspects: first, the maximum likelihood estimate of the noise is formulated according to a clear analysis of the probability distribution of the time-frequency coefficients. Second, the choice of an optimal minimal subset is investigated. The signal versus noise discrimination property of the spectral kurtosis is used to select a minimal subset which ensures a fair trade-off between the bias and the variance of the estimator. The resulting performances are discussed and compared with those of other methods through numerical simulations on synthetic signals. The use of the MiniSMaL estimator in a time-frequency detection procedure is finally illustrated on a real-world signal.