The Laguerre-Gaussian rotating cavity system is a special device for obtaining optomechanically induced transparency (OMIT). This paper further proposes performing nonlinear coupling and orbital angular momentum (OAM) exchange in this cavity to achieve the modulation for the third-order Kerr nonlinear effect. The analytical expressions of Kerr nonlinearities are obtained with the Hamiltonian model of the system. Numerical simulation reveals that when the system is modulated to be in the vicinity of the OMIT windows, the OAM-carrying optical field undergoes slight absorption and intense dispersion, which leads to the generation of giant Kerr nonlinear effects. In contrast to the conventional electromagnetically induced transparency (EIT) and OMIT, the proposed system can be modulated by OAM and other parameters to achieve giant Kerr nonlinearities. In addition, the orbital angular quantum number can also be used to control the group velocity of light propagation and achieve fast- and slow-light effects.