Using archival Chandra observations of 19 LINERs, we explore the X-ray properties of their inner kiloparsec to determine the origin of their nuclear X-ray emission, to investigate the presence of an AGN, and to identify the power source of the optical emission lines. The relative numbers of LINER types in our sample are similar to those in optical spectroscopic surveys. We find that diffuse, thermal emission is very common and is concentrated within the central few hundred parsecs. The average spectra of the hot gas in spiral and elliptical galaxies are very similar to those of normal galaxies. They can be fitted with a thermal plasma (kT ~ 0.5 keV) plus a power-law (photon index of 1.3-1.5) model. There are on average three detected point sources in their inner kiloparsec with 10³⁷ ergs s^-1 < L_{0.5-10 keV} < 10⁴⁰ ergs s^-1. The average cumulative luminosity functions for sources in spiral and elliptical galaxies are identical to those of normal galaxies. In the innermost circle of 25 radius in each galaxy we find an AGN in 12 of the 19 galaxies. The AGNs contribute a median of 60% of the 0.5-10 keV luminosity of the central 25 region, and they have luminosities of 10³⁷-10³⁹ ergs s^-1 (Eddington ratios of 10^-8 to 10^-5). The ionizing luminosity of the AGNs is not enough to power the observed optical emission lines in this particular sample. Thus, we suggest that the lines are powered either by the mechanical interaction of an AGN jet (or wind) with the circumnuclear gas or by stellar processes, such as photoionization by post-AGB stars from an old or intermediate-age population (in most cases) or by young stars (in a few cases).