= Solution
Put $c_i=\int_D\varphi_i(x)\,d\lambda_n(x)$. The stated scalar random variable is
$$
Z=\int_DU(x)\,d\lambda_n(x)
=\sum_{i=1}^k\sqrt{\nu_i}\,c_i\xi_i.
$$
As a finite <linear combination of independent normal random variables>, it is normally distributed. Its mean is zero and its variance is
$$
\boxed{\operatorname{Var}(Z)
=\frac12\sum_{i=1}^k\nu_i
\left(\int_D\varphi_i\,d\lambda_n\right)^2}.
$$
Finally, <orthonormal set>[orthonormality] of the $\varphi_i$ gives
$$
\|U\|_X^2=\sum_{i=1}^k\nu_i\xi_i^2.
$$
Since $\mathbb E\xi_i^2=1/2$,
$$
\boxed{\int_\Omega\|U\|_X^2\,d\mathbb P
=\frac12\sum_{i=1}^k\nu_i}.
$$
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