All 280 of the statistically complete Palomar sample of nearby (< 120 Mpc) galaxies {delta} > 20{deg} have been observed at 1.5 GHz as part of the LeMMINGs e-MERLIN legacy survey. Here, we present Chandra X-ray observations of the nuclei of 213 of these galaxies, including a statistically complete sub-set of 113 galaxies in the declination range 40{deg} < {delta} < 65{deg}. We observed galaxies of all optical spectral types, including 'active' galaxies [e.g. low-ionization nuclear emission line regions (LINERs) and Seyferts] and 'inactive' galaxies like H II galaxies and absorption line galaxies (ALG). The X-ray flux limit of our survey is 1.65*10^-14^ erg/s/cm^2^ (0.3-10 keV). We detect X-ray emission coincident within 2 arcsec of the nucleus in 150/213 galaxies, including 13/14 Seyferts, 68/77 LINERs, 13/22 ALGs and 56/100 H II galaxies, but cannot completely rule out contamination from non-AGN processes in sources with nuclear luminosities <~ 10^39^ erg/S. We construct an X-ray Luminosity function (XLF) and find that the local galaxy XLF, when including all active galactic nucleus (AGN) types, can be represented as a single power law of slope -0.54 +/- 0.06. The Eddington ratio of the Seyferts is usually 2-4 decades higher than that of the LINERs, ALGs, and H II galaxies, which are mostly detected with Eddington ratios <~ 10^-3^. Using [O III] line measurements and black hole masses from the literature, we show that LINERs, H II galaxies and ALGs follow similar correlations to low luminosities, suggesting that some 'inactive' galaxies may harbour AGN.