This paper proposes a two-layer robust fixed-time sliding mode control (FTSMC) scheme for the trajectory-tracking problem of a differential-drive LIMO mobile robot subject to modeling uncertainties and external disturbances. At the kinematic layer, a backstepping-based velocity command is designed to stabilize the posture error dynamics. At the dynamic layer, an integral-type nonsingular fixed-time sliding surface combined with a robust reaching law drives the velocity tracking errors to zero within a settling-time bound that is independent of the initial conditions. Unlike finite-time schemes, whose convergence time grows with the initial tracking error, the proposed method provides an explicit, pre-computable fixed-time guarantee while avoiding singularity and severe chattering. The main contribution is a rigorous fixed-time stability analysis with explicit settling-time bounds, together with a hardware-friendly control law that requires only the known upper bound of the lumped disturbance. Real-time experiments on the LIMO platform, conducted under both disturbance-free and disturbed conditions, confirm that the proposed FTSMC achieves the lowest Integral Squared Error (ISE), Integral Absolute Error (IAE), and Root Mean Square Error (RMSE) among the compared methods, reducing the ISE by up to a factor of 287, the IAE by up to a factor of 11, and the RMSE by up to a factor of 17. The findings demonstrate that the proposed scheme is a high-precision, hardware-friendly solution for real-time trajectory tracking of low-cost differential-drive mobile robots.