  In our investigation, we pioneer the development of geometrically deformed
strange stars within the framework of $f(\mathcal{T})$ gravity theory through
gravitational decoupling via the complete geometric deformation (CGD)
technique. The significant finding is the precise solution for deformed strange
star (SS) models achieved through the vanishing complexity factor scenario.
Further, we introduce the concept of space-time deformation caused by dark
matter (DM) content in DM haloes, leading to perturbations in the metric
potentials $g_{tt}$ and $g_{rr}$ components. Mathematically, this DM-induced
deformation is achieved through the CGD method, where the decoupling parameter
$\alpha$ governs the extent of DM influence. To validate our findings, we
compare our model predictions with observational constraints, including
GW190814 (with a mass range of $2.5-2.67 M_{\odot}$) and neutron stars (NSTRs)
such as EXO 1785-248 [mass=$1.3_{-0.2}^{+0.2}~M_{\odot}$], 4U 1608-52
[mass=$1.74_{-0.14}^{+0.14}~M_{\odot}$], and PSR J0952-0607
[mass=$2.35_{-0.17}^{+0.17}~M_{\odot}$]. Our investigation delves into the
stability of the model by considering causality conditions, Herrera's Cracking
Method, the adiabatic index, and the Harrison-Zeldovich-Novikov criterion. We
demonstrate that the developed model mimics a wide range of recently observed
pulsars. To emphasize its compatibility, we highlight the predicted mass and
radius in tabular form by varying both the parameters $\alpha$ and $\zeta_1$.
Notably, our findings are consistent with the observation of gravitational
waves from the first binary merger event. Furthermore, we compare our results
with those obtained for a slow-rotating configuration. In addition to this, we
discuss the moment of inertia using the Bejger-Haensel approach in this
formulation.
