= Solution
An <exponential family> has density $h(x)\exp\{\phi\cdot T(x)-k(\phi)\}$ on a parameter-independent support. For an <independent and identically distributed> sample, its <likelihood> factorizes through $\sum_iT(X_i)$, so the <Fisher-Neyman factorization theorem> gives a <sufficient statistic> whose dimension is the fixed dimension of $T$, independent of sample size.
Expanding the normal log-density gives $\phi_1=\mu/v$, $\phi_2=-1/(2v)$, and
$$
\boxed{k(\phi)=\frac12\log\frac\pi{-\phi_2}-\frac{\phi_1^2}{4\phi_2},\qquad\mathcal F=\{(\phi_1,\phi_2):\phi_1\in\mathbb R,\ \phi_2<0\}.}
$$
The <mean-value parameter> is $\nabla k=(\mathbb EX,\mathbb EX^2)$, hence
$$
\boxed{H_1=-\frac{\Phi_1}{2\Phi_2}=\mu,\qquad H_2=-\frac1{2\Phi_2}+\frac{\Phi_1^2}{4\Phi_2^2}=v+\mu^2.}
$$
For $n\geq2$ and positive sample variance, $\widehat\mu=\bar X$ and $\widehat v=n^{-1}\sum_i(X_i-\bar X)^2$, so $\widehat\Phi_1=\bar X/\widehat v$. For this <normal natural-mean parameter ratio estimator>, the <central limit theorem> gives asymptotic independent errors for $\widehat\mu$ and $\widehat v$, with variances $v/n$ and $2v^2/n$. Applying the <delta method> to $\mu/v$ yields
$$
\boxed{\sqrt n(\widehat\Phi_1-\Phi_1)\ \xrightarrow{d}\ N\left(0,\frac1v+\frac{2\mu^2}{v^2}\right).}
$$
If $v=v_0$ is known, $\Phi_2=-1/(2v_0)$ is fixed and $\widehat\Phi_1=\bar X/v_0$ has exact distribution $\boxed{N(\Phi_1,1/(nv_0))}$. The variance-estimation contribution $2\mu^2/v^2$ disappears.
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