Final answer:
The nurse should observe and document peeling skin on fingers as a characteristic sign or symptom of Kawasaki's disease in a child.
Step-by-step explanation:
A characteristic sign or symptom that the nurse should observe and document in a child admitted with possible Kawasaki's disease is peeling skin on fingers. Kawasaki's disease is an inflammatory condition that affects the blood vessels, and one of the hallmark features of the disease is skin peeling, typically on the fingers and toes. This symptom, along with other criteria, can help in confirming the diagnosis of Kawasaki's disease.