To determine which city was part of the Byzantine Empire the longest, let's examine the information given for each city:
- Rome: Part of the Byzantine Empire around the year 550 but not mentioned in later years.
- Athens: Part of the Byzantine Empire around 550, remained part in 1020, and was not specifically excluded in 1360.
- Jerusalem: Part of the Byzantine Empire around 550 but not mentioned in later years.
- Kiev: The city is not mentioned in any of the time frames provided, indicating it was not part of the Byzantine Empire.
Based on the information, Athens is the only city that is confirmed to have been part of the Byzantine Empire from 550 through at least 1020, making it the city that was part of the empire the longest.
Answer: Athens