In the present work, we analyzed the Kantowski-Sachs cosmological model and teleparallel gravity, where a bulk viscous fluid containing one-dimensional cosmic strings is the source for the energy−momentum tensor. To obtain the deterministic solution of the field equations, we employed the proportionality condition linking the shear scalar \((\sigma )\) and the expansion scalar \((\theta )\) , establishing a relationship between metric potentials. Another approach employed is the hybrid expansion law (HEL). The discussion focuses on the behavior of the accelerating universe concerning the specific choice of a nonlinear (or power law model) of teleparallel gravity \(f(T)=\alpha T + \beta T^m\) , where T is the torsion scalar, \(\alpha\) , and \(\beta\) are model parameters and m is restricted to greater than or equal to 2. The effective equation of the state parameter \((\omega _{eff})\) of models will support the acceleration of the universe. We observed that the null and weak energy conditions are obeyed but violate the strong energy condition as per the present accelerating expansion. Under specific model parameter constraints, the universe shows a transition from a decelerating to an accelerating phase.
