The purpose of structural health monitoring (SHM) for structural assessment and damage identification is based on dynamic response data collected by monitoring systems. However, economic constraints and the complexity of civil structures prevent monitoring systems from acquiring complete structural response information, which hinders accurate structural assessment with sparse sensors. Therefore, response reconstruction of all degrees of freedom from limited measurement data will significantly enhance SHM performance. The Kalman filtering (KF) is commonly used for response reconstruction, which provides optimal solution for systems well-represented by a fully known Gaussian linear state space model. However, the KF assumes that measurement and process noise follow a known white Gaussian noise distribution, which is impractical in many civil engineering applications considering the variations of environmental conditions and non-negligible modeling errors. To overcome this challenge, a hybrid modeling technique, called KalmanNet, is applied in this study for state estimation of partially known systems. The KalmanNet integrates a recurrent neural network (RNN) module into the process of KF, which replaces the computation process of Kalman gain. The RNN learns to compute Kalman gain from actual monitoring data under the influence of uncertain noise distribution and modeling errors, without necessitating any Gaussian assumptions or noise covariance specifications. A three-story shear-wall structure is taken as a numerical study where different types of noise and modeling errors are studied. The results show that the KalmanNet can effectively reconstruct structural response from sparse measurements in real time with high accuracy under the influence of non-Gaussian noise and modeling errors.