In order to solve the problem of axisymmetric indentation of an incompressible elastic film on a rigid substrate, a simple analytical method based on Kerr-model is derived, in which the differential relation between the contact pressure and the displacement of the film's upper surface is established. Then, the high-order asymptotic solution to the problem is solved by using Betti's reciprocal theorem and the explicit relation between contact pressure, indentation depth and contact radius is built. When the high-order term is ignored, the present asymptotic solution is the same as the existing low-order solution. In addition, a finite element model is established to verify the accuracy of the asymptotic solution. The result shows that, compared with the existing low-order asymptotic solutions, the higher-order asymptotic solution agrees better with the existing numerical results and the newly developed numerical simulation results.