The question asks two things: (1) how to show that the maximum $X_{(n)}$ converges, in the sense that $(X_{(n)}-b_n)/a_n$ converges (in distribution) for suitably chosen sequences $(a_n)$ and $(b_n)$, to the Standard Gumbel distribution and (2) how to find such sequences.
The first is well-known and documented in the original papers on the Fisher-Tippett-Gnedenko theorem (FTG). The second appears to be more difficult; that is the issue addressed here.
Please note, to clarify some assertions appearing elsewhere in this thread, that
The maximum does not converge to anything: it diverges (albeit extremely slowly).
There appear to be different conventions concerning the Gumbel distribution. I will adopt the convention that the CDF of a reversed Gumbel distribution is, up to scale and location, given by $1-\exp(-\exp(x))$. A suitably standardized maximum of iid Normal variates converges to a reversed Gumbel distribution.
Intuition
When the $X_i$ are iid with common distribution function $F$, the distribution of the maximum $X_{(n)}$ is
$$F_n(x) = \Pr(X_{(n)}\le x) = \Pr(X_1 \le x)\Pr(X_2 \le x) \cdots \Pr(X_n \le x) = F^n(x).$$
When the support of $F$ has no upper bound, as with a Normal distribution, the sequence of functions $F^n$ marches forever to the right without limit:

Partial graphs of $F_n$ for $n=1,2,2^2, 2^4, 2^8, 2^{16}$ are shown.
To study the shapes of these distributions, we can shift each one back to the left by some amount $b_n$ and rescale it by $a_n$ to make them comparable.

Each of the previous graphs has been shifted to place its median at $0$ and to make its interquartile range of unit length.
FTG asserts that sequences $(a_n)$ and $(b_n)$ can be chosen so that these distribution functions converge pointwise at every $x$ to some extreme value distribution, up to scale and location. When $F$ is a Normal distribution, the particular limiting extreme value distribution is a reversed Gumbel, up to location and scale.
Solution
It is tempting to emulate the Central Limit Theorem by standardizing $F_n$ to have unit mean and unit variance. This is inappropriate, though, in part because FTG applies even to (continuous) distributions that have no first or second moments. Instead, use a percentile (such as the median) to determine the location and a difference of percentiles (such as the IQR) to determine the spread. (This general approach should succeed in finding $a_n$ and $b_n$ for any continuous distribution.)
For the standard Normal distribution, this turns out to be easy! Let $0 \lt q \lt 1$. A quantile of $F_n$ corresponding to $q$ is any value $x_q$ for which $F_n(x_q) = q$. Recalling the definition of $F_n(x) = F^n(x)$, the solution is
$$x_{q;n} = F^{-1}(q^{1/n}).$$
Therefore we may set
$$b_n = x_{1/2;n},\ a_n = x_{3/4;n} - x_{1/4;n};\ G_n(x) = F_n(a_n x + b_n).$$
Because, by construction, the median of $G_n$ is $0$ and its IQR is $1$, the median of the limiting value of $G_n$ (which is some version of a reversed Gumbel) must be $0$ and its IQR must be $1$. Let the scale parameter be $\beta$ and the location parameter be $\alpha$. Since the median is $\alpha + \beta \log\log(2)$ and the IQR is readily found to be $\beta(\log\log(4) - \log\log(4/3))$, the parameters must be
$$\alpha = \frac{\log\log 2}{\log\log(4/3) - \log\log(4)};\ \beta = \frac{1}{\log\log(4) - \log\log(4/3)}.$$
It is not necessary for $a_n$ and $b_n$ to be exactly these values: they need only approximate them, provided the limit of $G_n$ is still this reversed Gumbel distribution. Straightforward (but tedious) analysis for a standard normal $F$ indicates that the approximations
$$a_n^\prime = \frac{\log \left(\left(4 \log^2(2)\right)/\left(\log^2\left(\frac{4}{3}\right)\right)\right)}{2\sqrt{2\log (n)}},\ b_n^\prime = \sqrt{2\log (n)}-\frac{\log (\log (n))+\log \left(4 \pi \log ^2(2)\right)}{2 \sqrt{2\log (n)}}$$
will work fine (and are as simple as possible).

The light blue curves are partial graphs of $G_n$ for $n=2, 2^6, 2^{11}, 2^{16}$ using the approximate sequences $a_n^\prime$ and $b_n^\prime$. The dark red line graphs the reversed Gumbel distribution with parameters $\alpha$ and $\beta$. The convergence is clear (although the rate of convergence for negative $x$ is noticeably slower).
References
B. V. Gnedenko, On The Limiting Distribution of the Maximum Term in a Random Series. In Kotz and Johnson, Breakthroughs in Statistics Volume I: Foundations and Basic Theory, Springer, 1992. Translated by Norman Johnson.