Do you understand that you do not need to, and, indeed, are not expected to actually find the image and kernel in order to do this? You are told that $\displaystyle T^2(v)= T(T(v))= 0$ for every v in the vector space. There are an infinite number of functions, T, that will do that and they have different images and kernels. In order to prove that, for such a function, $\displaystyle Im(T)\subseteq Ker(T)$ you just need to know the definitions of "image" and "kernel".