  Employing $f(T)$ gravity, where $T$ is the torson, we have developed a new
model of an anisotropic compact star in this work. Tolman-Kuchowicz (TK) metric
potential has been used to solve the set of field equations. Furthermore, the
matching conditions for interior and exterior geometry have been discussed. We
have considered observation data of the compact star LMC X-4 and analyzed
thermodynamical properties (density, pressure, equation of state parameter,
square speed of sound, and equilibrium condition) analytically and graphically
to test the validity of the solution. The compact star is found to meet the
energy conditions. Through the causality condition and Herrera's cracking
concept, the stability analysis of the present model has been presented and it
confirms the physical acceptability of the solution. It has been shown that the
obtained interior solutions for compact stars are consistent with all necessary
physical criterions and therefore relevant as well as physically acceptable.
