Modeling the complex flow behavior of impingement jets is a problem of great importance in many industrial applications. Traditional modeling methods often fail to accurately predict these flows due to their nonlinear nature. This paper presents a neural network-based reduced-order model for experimental data of a circular impinging jet and compares several data assimilation frameworks for incorporating wall shear stress measurements obtained from different radial positions. The high-dimensional velocity field and the corresponding wall shear stress measurements are obtained using time-resolved particle image velocimetry and polarographic measurements, respectively. The developed reduced-order model results from a proper orthogonal decomposition (POD) step for dimensionality reduction with a neural ordinary differential equation (NODE) for temporal modeling. The performance of the POD-NODE framework is compared with dynamic mode decomposition and nonlinear temporal modeling using long short-term memory. Assessments are based on root mean squared error and spectral proper orthogonal decomposition of the reconstructed predicted solution. It is found that the POD-NODE framework provides the most accurate dynamical model. Furthermore, it is evident that incorporating wall shear stress measurements in the NODE model as additional states significantly improves the prediction accuracy, outperforming traditional filtering techniques such as extended Kalman filters.