Combine kumquats and water in a blender and puree until mostly smooth. Pour the mixture through a fine-mesh sieve into a medium saucepan, pressing on the solids to extract as much liquid as possible. Add sugar and bring to a boil over high heat. Reduce heat to a simmer and cook, stirring occasionally, until thick and syrupy, 20 to 25 minutes. Let cool to room temperature, about 15 minutes, then refrigerate until cold, about 1 hour.