In this paper, we refine our method for the abundance analysis of high-resolution spectroscopy of the integrated light of unresolved globular clusters (GCs). This method was previously demonstrated for the analysis of old (>10Gyr) Milky Way (MW) GCs. Here, we extend the technique to young clusters using a training set of nine GCs in the Large Magellanic Cloud. Depending on the signal-to-noise ratio of the data, we use 20-100 Fe lines per cluster to successfully constrain the ages of old clusters to within a ~5Gyr range, the ages of ~2Gyr clusters to a 1-2Gyr range, and the ages of the youngest clusters (0.05-1Gyr) to a ~200Myr range. We also demonstrate that we can measure [Fe/H] in clusters with any age less than 12Gyr with similar or only slightly larger uncertainties (0.1-0.25dex) than those obtained for old MW GCs (0.1dex); the slightly larger uncertainties are due to the rapid evolution in stellar populations at these ages. In this paper, we present only Fe abundances and ages. For several of the clusters in this sample, there are no high-resolution abundances in the literature from individual member stars; our results are the first detailed chemical abundances available.