Finding the singularity order of a nonlinear ordinary differential equation (ODE) can be a challenging task, and there is no specific method applicable to all cases. However, I can provide you with some general guidance on how to approach this problem.

1. Review the Literature: Start by reviewing the existing literature on the Kurdyashov simplest method and its application to nonlinear ODEs. Look for examples or explanations that demonstrate how to determine the singularity order using this method. The literature may provide insights into specific techniques or considerations that can help you in your analysis.

2. Analyze the Equation: Examine the ODE you are working with, particularly the polynomial form of G(y, y', y''). Try to identify any singular points or regions where the equation may become undefined or have unusual behavior. Singular points often occur when certain terms or factors in the equation become zero or infinite.

3. Singular Points and Behavior: Investigate the behavior of the equation near the identified singular points. Determine whether the singularity is removable (the equation can be regularized at the singularity) or essential (the equation remains singular at the point).

4. Analytical Techniques: Consider employing analytical techniques such as series expansions, power series, or transformation methods to study the behavior of the equation near the singularity points. These techniques may help in determining the nature and order of the singularity.

5. Numerical Methods: If the analytical techniques are not yielding conclusive results, you may resort to numerical methods. Numerical methods can provide insights into the behavior of the equation near singular points by approximating the solutions. Tools like numerical integration or software packages specifically designed for solving ODEs can assist in this process.

It's important to note that the singularity order determination can be a complex task and may require expertise in nonlinear dynamics and differential equations. If you are encountering difficulties, it would be beneficial to consult with a mathematician or an expert in the field who can provide further guidance and assistance tailored to your specific problem.