Control of spontaneous emission in a microcavity has many important applications, e.g. improvement of the efficiency of light emitting devices. InAs quantum dots (QDs) embedded in microdisks are ideal systems for spontaneous emission control. The whispering gallery (WG) modes of microdisks have low volume and high quality factor. The homogeneous linewidth of InAs quantum dots is smaller than the spectral width of WG modes. Thus, a large enhancement of the spontaneous emission rates should be expected for QDs coupled to WG modes. However, large inhomogeneous broadening of the QD energy levels and random spatial distribution of the QDs in a microdisk lead to a broad distribution of the spontaneous emission rates. Using an efficient regularized method based on the truncated singular value decomposition and the non-negative constraints, we extract the distribution of spontaneous emission rates from the temporal decay of emission intensity. The maximum spontaneous emission enhancement factor exceeds 10.