= Solution
The part of the <log-likelihood> depending on $\theta$ is
$$
\ell(\theta)=-\frac12\sum_{i=1}^n(X_i-\theta)^T\Sigma^{-1}(X_i-\theta)+\text{constant}.
$$
<Completing the square> separates the sum into a term independent of $\theta$ and $n(\overline X-\theta)^T\Sigma^{-1}(\overline X-\theta)$. Positive definiteness makes its unique minimum occur at $\overline X$. Since linear combinations of independent <multivariate normal> vectors are normal,
$$
\boxed{\widehat\theta_n=\overline X,\qquad\widehat\theta_n\sim\mathcal N_d(\theta,\Sigma/n).}
$$
The <Hessian matrix> $-n\Sigma^{-1}$ is negative definite, confirming the global likelihood maximum.
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