This work introduces a general theoretical framework, which expresses the topological invariants of two-dimensional Floquet systems in terms of tractable response functions: Building on the Sambe representation of periodically-driven systems, and inspired by the Středa formula for static systems, we evaluate the flow of the unbounded Floquet density of states in response to a magnetic perturbation. This Floquet-Středa response, which is a priori mathematically ill-defined, is regularized by means of a Cesàro summation method. As a key outcome of this approach, we relate all relevant Floquet winding numbers to simple band properties of the Floquet-Bloch Hamiltonian. These general relations indicate how the topological characterization of Floquet systems can be entirely deduced from the stroboscopic time-evolution of the driven system. Importantly, we identify two physically distinguishable contributions to the Floquet-Středa response: a quantized flow of charge between the edge and the bulk of the system, and an 'anomalous' quantized flow of energy between the system and the driving field, which provides new insight on the physical origin of the anomalous edge states. As byproducts, our theory provides: a general relation between Floquet winding numbers and the orbital magnetization of Floquet-Bloch bands; a local marker for Floquet winding numbers, which allows to access Floquet topology in inhomogeneous samples; an experimental protocol to extract these Floquet winding numbers from density-measurements in the presence of an engineered bath; as well as general expressions for these topological invariants in terms of the magnetic response of the Floquet density of states, opening a route for the topological characterization of interacting Floquet systems.