This contribution presents a novel approach for the calibration of distributed parameters in a 2D urban flood hydraulic model. It focuses on the challenging issue of inferring distributed friction parameters from multi-source heterogeneous spatio-temporal observations of their hydraulic signatures in the context of an urban flash flood in a complex street network. A variational data assimilation algorithm is used to infer high-dimensional multi-variate parameters (spatialized friction and inflow discharge time series) using multi-source observations. This method relies on a differentiable 2D shallow water hydraulic model which enables to generate high-resolution sensitivity maps of local gradients and Derivative-based Global Sensitivity Measures (DGSM), enabling to guide adequate definition of parameter spatialization for the data assimilation process. Assimilated data include real local limnigraphic measurements and high-water marks collected after a major flood event, as well as modeled flow velocity used in twin experiments setups. This study is the first to leverage high-water marks with a variational method for the calibration of distributed parameters in an urban flood model. The multi-source data is used to infer inflow hydrographs and distributed friction parameters in setups of varying complexity. In the main setup, the complex structure of the street network, along with the sensitivity maps and hydraulics expertise led to define a model configuration with 45 friction patches. A high-dimensional parameter vector composed of these friction values and an upstream inflow is inferred simultaneously by assimilating real limnigraphic data and high-water marks. This leads to an increase in model fit to observations and satisfying parameter estimates.