We investigate the dynamics of a kicked particle in an infinite square well undergoing frequent measurements of energy. For a large class of periodic kicking forces, constant diffusion is found in such a non-Kolmogorov-Arnol'd-Moser system. The influence of a phase shift of the kicking potential on the short-time dynamical behavior is discussed. The general asymptotical measurement-assisted diffusion rate is obtained. The entanglement between the particle and the measuring apparatus is investigated. There exist two distinct dynamical behaviors of entanglement. The bipartite entanglement between the system of interest and the whole spin of the measuring apparatus grows with the kicking steps and it gains a larger value for a more chaotic system. However, the partial entanglement between the system of interest and the partial spin of the measuring apparatus decreases with the kicking steps. The relation between the entanglement and quantum diffusion is also analyzed.