= Solution
Enumerate the <orthonormal basis> consisting of $H_0$ and the <Haar wavelets> $H_{n,k}$ as $(e_j)_{j\geq1}$, and enumerate the corresponding <independent> <standard normal random variables> as $(A_j)_{j\geq1}$. Set $c_j=\langle h,e_j\rangle_{\mathcal H}$. The <Parseval identity for a Hilbertian basis> gives
$$
\sum_{j=1}^\infty c_j^2=\|h\|_{\mathcal H}^2<\infty.
$$
The partial sums $T_m=\sum_{j=1}^mc_jA_j$ form a <martingale> for $\sigma(A_1,\ldots,A_m)$, since the next summand is <independent> of the past and has <mean> zero. Moreover,
$$
\mathbb ET_m^2=\sum_{j=1}^mc_j^2\leq\|h\|_{\mathcal H}^2.
$$
The <martingale convergence theorem> for a <martingale> bounded in <L2 space> proves convergence <almost surely> and <convergence in L2>. Equivalently one may apply the <Kolmogorov convergence theorem> to the summable <variances> $c_j^2$.
A finite <linear combination of independent normal random variables> is normal. Its <characteristic function> is $\mathbb E e^{iuT_m}=\exp(-u^2\sum_{j\leq m}c_j^2/2)$. Passing to the limit identifies
$$
\boxed{Wh=\lim_{m\to\infty}T_m\quad\text{almost surely and in }L^2(\mathbb P),
\qquad Wh\sim N(0,\|h\|_{\mathcal H}^2).}
$$
For $h=0$ this is the constant zero <random variable>. Each fixed $h$ has its own permissible null set; simultaneous pathwise convergence for all $h\in\mathcal H$ is unnecessary.
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