The paper presents a continuous time-varying feedback control law for the stabilization of a class of driftless nonholonomic systems to a given configuration. Two-input systems that can be converted in chained form include wheeled mobile robots (e.g., the \(N\)-trailer system driven by a car) as well as other kinematic systems with Pfaffian nonholonomic constraints. This paper takes advantage of the latest developments in the field of time-varying feedback controllers for nonlinear systems for which no smooth time-invariant stabilizing feedback exists. The continuous, although non-smooth, control law is the result of weighting with an autonomous time-varying function \(f(t)\) of a discontinuous function of the state which is forced to switch only at instants where \(f(t)\) is zero. Although the analysis is quite involved, the proposed control law is rather simple to implement. The closed-loop system is shown to be \(K\)-exponentially stable, a relaxed notion of exponential stability. It should be noted that this guarantees an exponential rate of convergence to zero for the error, but does not imply the typical robustness properties induced by an exponentially stable feedback controller. The authors work directly with the chained form of a nonholonomic system, so that the singularities that are often present in the transformation to this form are not taken into account. Therefore, the globality of the obtained results should be considered carefully in real applications.