Je ne sais pas que faire ici.

English Translation

I don’t know what to do here.

Wouldn’t it be Je ne sais pas quoi faire ici, and not que faire?

Beauty is in the eye of the beholder.

I suspect that your suggestion is more grammatical. However, if you were hanging with your buds down the local bar your buds would think you were an ass for speaking like that. Alternatively, if you were trying to impress your boss things would be different.

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I think both phrases are correct and common.
Possibly ‘que’ would be more frequent when the negation ‘pas’ is dropped, but to my (non-native fluent) ears, they are more or less equivalent.
Contrary to what was suggested, I think ‘que’ might be considered a more literary form.

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