All 280 of the statistically-complete Palomar sample of nearby (<120 Mpc) galaxies dec > 20 degrees 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 degrees to 65 degrees. We observed galaxies of all optical spectral types, including 'active' galaxies (e.g., LINERs and Seyferts) and 'inactive' galaxies like HII galaxies and absorption line galaxies (ALG). The X-ray flux limit of our survey is 1.65$\times$10$^{-14}$ergs$^{-1}$ cm$\mathrm{<mrow\; data-mjx-texclass="ORD">-2</mrow>}$ (0.3$-^{39}$ erg s$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$. We construct an X-ray Luminosity function (XLF) and find that the local galaxy XLF, when including all AGN types, can be represented as a single power-law of slope $-0.54\pm 0.06$. The Eddington ratio of the Seyferts is usually 2-4 decades higher than that of the LINERs, ALGs and HII galaxies, which are mostly detected with Eddington ratios <10$\mathrm{<mrow\; data-mjx-texclass="ORD">-3</mrow>}$. Using [O III] line measurements and BH masses from the literature, we show that LINERs, HII galaxies and ALGs follow similar correlations to low luminosities, suggesting that some 'inactive' galaxies may harbour AGN.