The word that best completes the sentence is "defiant."
The sentence would read: "Annika tried to get her little brother to bed, but he was defiant."
The reason "defiant" is the correct choice is that it is an adjective that describes her little brother's behavior, indicating that he is resisting or opposing Annika's attempt to get him to bed. The other options do not fit grammatically in the sentence: "defiance" is a noun, "defy" is a verb, and "defiantly" is an adverb.