All 280 of the statistically complete Palomar sample of nearby (<120 Mpc) galaxies δ > 20° 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° <δ < 65°. 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 ${\rm H\, \small {II}}$ galaxies and absorption line galaxies (ALG). The X-ray flux limit of our survey is 1.65 × 10-14 erg s-1 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 ${\rm H\, \small {II}}$ galaxies, but cannot completely rule out contamination from non-AGN processes in sources with nuclear luminosities $\lesssim 10^{39}$ erg s-1. 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 ${\rm H\, \small {II}}$ galaxies, which are mostly detected with Eddington ratios $\lesssim 10^{-3}$. Using [${\rm O\, \small {III}}$] line measurements and black hole masses from the literature, we show that LINERs, ${\rm H\, \small {II}}$ galaxies and ALGs follow similar correlations to low luminosities, suggesting that some 'inactive' galaxies may harbour AGN.