We present an X-ray spectroscopic analysis of two active dwarf stars with extremely short rotation periods, HD 197890 (Speedy Mic) and Gliese 890, using X-ray data acquired with the ASCA observatory. We analyze the X-ray spectrum of Speedy Mic in two separate time intervals, one corresponding to an apparent stellar flare, the other to "quiescence" or the nonflaring state. We also present a reanalysis of the ASCA spectrum of the M dwarf stellar binary YY Gem, during both quiescent and flaring states. We use the recently updated MEKAL plasma code to model these spectra and find that a minimum of two temperature components are required to obtain good fits for all three stars. The inferred emission measures of the hot components in YY Gem and Speedy Mic, and the temperature of the hot component in the case of YY Gem, increase significantly during the observed X-ray flares, in agreement with previous studies of similar stellar X-ray flares. In addition, metal abundances that are low compared to the solar photospheric values are preferred (but not required) for the quiescent coronae of Gliese 890 and Speedy Mic, while for the quiescent corona of YY Gem, subsolar metal abundances are required and solar abundance models can be ruled out with high confidence. We combine the results of this study with those of previous ASCA and ROSAT analyses of late-type dwarf stars with known rotation rates to obtain a sample of 17 G, K, and M dwarfs. We find a strong correlation of the X-ray to bolometric luminosity ratio, L X /L bol , both with the rotation period and the Rossby number (R o ), in agreement with previous studies. Using a homogeneous subset of 10 dwarfs for which we have purely ASCA-derived temperatures and elemental abundances, we find that the temperature of the hottest component present in their coronae is correlated with this luminosity ratio, as expected. Finally, we note that the Fe abundances for all active dwarf stars having normalized X-ray luminosities log (L X /L bol ) > −3.7 are significantly subsolar, whereas the Fe abundances in the less active stars are within a factor of 2 of the solar value.