Sylow 2-subgroups of \(D_{2n}\) (D&F 4.5.12)

Let \(2n = 2^a k\) where \(k\) is odd. Prove that the number of Sylow 2-subgroups of \(D_{2n}\) is \(k\).

One approach is to prove that \(N_{D_{2n}}(P) = P\) for any \(P \in Syl_2(D_{2n})\) and calculate \(n_2\) directly. However I have tried for hours to work out this approach to no avail. So here is an alternative approach:

There are \(k\) Sylow 2-subgroups of the form \(\langle sr^i, r^k \rangle\), where \(0 \le i \le k-1\), so \(n_2 \ge k\). Furthermore, \(n_2 \mid k\) so \(n_2 \le k\).