Combine water and mint leaves in a medium bowl; cover and let stand 10 minutes. Strain mixture through a fine sieve; discard mint. Pour liquid evenly into 28 ice cube tray compartments; freeze until firm.
Combine first 4 ingredients in a medium bowl; cover and steep 10 minutes. Discard tea bags. Strain mixture through a fine sieve; discard mint and tea leaves. Stir lime juice and honey into tea.
Serve tea over Mint Ice Cubes. Garnish with mint sprigs, if desired.