Finding a UMVUE for variance of normal distribution

#1
Let Let \(X_1,X_2,...,X_n\) be a random sample from a normal distribution with mean \(\mu \)and variance \(\sigma^2 \).
I showed that \((\bar X,S^2) \) is jointly sufficient for estimating ( \(\mu \), \(\sigma^2 \)) where \(\bar X \)is the sample mean and \(S^2 \) is the sample variance.

Then assuming that \((\bar X,S^2) \) is also complete I have to show that \(\sqrt{ n-1\over 2}{\Gamma ({ n-1\over 2})\over\Gamma (\frac n2)} S \)
is a Uniformly Minimum Variance Unbiased Estimator for \(\sigma \).

I think I have to use Lehman Scheffe theorem as \((\bar X,S^2) \)is jointly sufficient and complete for \(\sigma \).
But how can I find a function which is unbiased for \(\sigma \) that contains both \((\bar X,S^2) \).

I don't understand how to work when there's a joint sufficiency and completeness.