Gyrochronology, the method of inferring the age of a star from its rotation period, could provide ages for millions or even billions of stars in the upcoming time-domain era. However, the relationship between stellar age and rotation rate is still poorly calibrated for low-mass K and M dwarf stars. Few methods are effective at providing ages for individual low-mass stars and a suitable calibration sample therefore remains elusive. However, kinematic ages, which can be calculated for ensembles of stars, are generally applicable to stars of all masses and could provide a suitable calibration sample for gyrochronology. We have estimated the kinematic ages of thousands of Kepler FGKM dwarfs and subgiants by calculating the velocity dispersions of populations of stars with similar rotation periods, temperatures, and luminosities (and assumed similar age). We then convert velocity dispersion to kinematic age via an age-velocity dispersion relation. We demonstrate the precision and accuracy of kinematic ages calculated this way by comparing them to ages independently determined via asteroseismology and isochrone fitting. We present new insights into the rotational evolution of stars, revealed by this kinematic age analysis.