It might help to start with a concrete example of a function, like $f(x,y)=10^9 x^2+y^2$ (the example they use on p.26). I'm trying to guess what went wrong with your analysis, because you haven't given much information.
I am mostly struggling with cross-terms of hessian.
$$f_1(x,y) = 10^9 x^2+y^2$$
and its rotation by $\theta=\frac\pi4$
$$f_2(x,y) = 10^9 (x\cos\theta + y\sin\theta)^2 + (-x\sin\theta + y\cos\theta)^2.$$
These are equivalent, and both are poorly scaled, but in different directions. If you try to analyze the Hessian in a way that just assumes that the function is poorly scaled specifically in the coordinate directions (only the diagonal terms are important, and so you don't care about cross terms), it's not going to work for the second function. I think you just need a different way of proving ill-conditionedness of the Hessian.