What are the challenges of using gradient descent on hyper parameters λ and η to find out their optimum values?
The cost function $C(w,\lambda)$ is not a function of learning rate $\eta$. You can't compute the gradient of $\eta$.

$\lambda$ is part of the cost function. You can indeed compute its gradient and apply SGD on $\lambda$. However, when you do that, $\lambda$ will keep decreasing because its optimal value is obviously $-\infty$.

Note that it is possible to optimize hyperparameters like $\eta$ and $\lambda$ with another neural net, but that is a different story.