Final answer:
The word that best completes the sentence by expressing the cheerful and eager demeanor with which Eliza repairs her snare is 'alacrity'.
Step-by-step explanation:
The correct word to fill in the blank in the sentence "and would repair afresh with cheerful _______ the damage done to her snare by the unwelcome visitor" is alacrity. Alacrity means cheerfulness or eagerness, and in the context of the given sentence, it indicates that Eliza approaches the task of repairing her snare with a positive and eager attitude. Words like foresight, nurture, and confines do not fit the context as neatly as alacrity does.