We use the quasienergy structure emerging in a periodically driven fluxonium superconducting circuit to encode quantum information with dynamically induced flux-insensitive sweet spots. The framework of Floquet theory provides an intuitive description of these high-coherence working points located away from the half-flux symmetry point of the undriven qubit. This approach offers flexibility in choosing the flux bias point and the energy of the logical qubit states as shown in Huang et al.[arXiv:2004.12458 (2020)]. We characterize the response of the system to noise in the modulation amplitude and dc flux bias, and experimentally demonstrate an optimal working point that is simultaneously insensitive against fluctuations in both. We observe a 40-fold enhancement of the qubit coherence times measured with Ramsey-type interferometry at the dynamical sweet spot compared with static operation at the same bias point.