You need to use Siegel's theorem (which is quite deep and relies on Roth's theorem or some such). This is in Chapter IX of *Arithmetic of Elliptic Curves*, specifically Theoream IX.3.1. You'll need to unsort the definitions a bit, since it's stated for number fields, but in your notation, one has
$$
\lim_{n\to\infty} \frac{2\log d_n}{n^2}
= \lim_{n\to\infty} \frac{\log |x_n|}{n^2}
= \hat h(P).
$$
Hmmm.. Actually, the definitions are unsorted for you in Example IX.3.3, where you'll find the following formula (using your notation) in the middle of page 279 (of the 2nd edition):
$$ \lim_{n\to\infty} \frac{\log|x_n|}{\log d_n^2} = 1. $$

BTW, the sequence $(d_n)_{n\ge1}$ is called the *Elliptic Divisibility Sequence* associated to the curve $E$ and point $P$. The fact that $\log d_n$ grows like a multiple of $n^2$ is an essential fact used to prove that elliptic divisibility sequences satisfy the Zsigmondy property: for all but finitely many $n$, there is a prime $p$ such that $p\mid d_n$ and $p\nmid d_m$ for all $m < n$.