I can't understand the meaning of the phrase "I'll have it nicely dried".
Does it mean the same as "I'll dry it very well"?
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Chambers gives a definition of nice as done with great care and exactness and I think the element of 'care' is relevant here. The person will be taking care to ensure that whatever it is is dried and ready for use.
Nicely ( or the variant of nice and...) personalises what would otherwise be a bald statement of efficiency.
It may not be you but some other person or the nature (wind, sun) that dries the thing.
You could have it dried by your mother, for example.