The mirror dark matter model accommodates dark matter in a hidden sector which is isomorphic to the Standard Model, containing mirror partners of Standard Model particles. A weak kinetic mixing interaction allowed by the symmetry between the two sectors gives interactions between charged ordinary and mirror particles. The LUX dark matter direct detection experiment would be sensitive to electron recoils from interactions of incoming mirror electrons with atomic electrons in the liquid xenon. The model for incoming mirror electrons is complicated by shielding effects from captured mirror dark matter within the Earth. Here the first calculations of the mirror electron scattering rate in liquid xenon, accounting for shielding, are carried out. The first direct detection search for mirror dark matter is then conducted, using 95 days of LUX data from 2013. With no evidence for an electron recoil signal, we rule out temperatures above 0.3 keV within the assumptions made for this model. Below this temperature we place a 90% confidence limit on the kinetic mixing parameter.