She —— wash the plates in the kitchen leaving her study.

The correct verb to complete the sentence "She _____ wash the plates in the kitchen leaving her study" is needs not.

Therefore, the complete sentence is: "She needs not wash the plates in the kitchen leaving her study."

This sentence implies that she is not required or obligated to wash the plates, possibly because she has other priorities or someone else can do it.

The other options, "needs nots" and "need not to," are incorrect grammatical forms.