One day in advance, combine the rice, crumbled cinnamon, and hot water in a bowl. Cool, cover, and set aside. Refrigerate if the weather is hot.
The next day, stir the nuts and coconut milk into the soaked rice. Working in batches, add the rice mixture to a blender and process until quite smooth. Each batch should take about 5 minutes. Strain through a medium-mesh sieve and pour into a pitcher.
In a saucepan over low heat, combine the sugar, cold water, and lime zest. Cook stirring, until the sugar is just dissolved. Let cool. Then remove the lime zest. Pour the cooled syrup into the pitcher and mix well. If the drink is too thick for your taste, add more water. Cover and refrigerate until chilled.
Stir well and pour over ice cubes in tall glasses. If desired, add a splash of rum to each glass before adding the horchata. Garnish with a mint sprig and a piece of cinnamon bark.