Grind mint leaves, ginger, and cardamom to a paste with a mortar and pestle, or mince them in a food processor. Steep the pulp in the hot water for half an hour. Strain the mixture through a cloth or sieve, collect the juice, and discard the pulp.
Blend mint and ginger juice and honey in a large bowl. Add ice, lemon juice, and orange juice. Serve in chilled glasses garnished with an orange ring and mint leaves.