Generalization of Banach's fixed point theorem

I think your answer is fine, but personally, I would do this:

$f^n$ is a contraction map, with constant $K$, hence has a fixed point $p$

For a fixed $x$, let $x_1=f(x), x_2=f^2(x), ...$ and $x_n=f^n(x)$

$d(f^N(x), f^N(p)) \leq K^m d(p, f^l(x))=K^m d(p, x_l)\leq K^m\max\limits_{i=1}^n d(x_i,p) $. Here $l$ is the smallest positive integers such that $N=m n+l$ now, as $N\rightarrow \infty$, so does $m$, $K^m\max\limits_{i=1}^n d(x_i,l)\rightarrow 0$

EDIT: I realise, I didn't give a proof that $p$ is a fixed point proof of $f$, but your proof of this is how I would do it too.