Honeydew, Jicama and Cucumber Sangria

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1/2 jicama, peeled and cut into 1/4-by-3/4-inch strips

1/4 honeydew melon, seeds and skin removed, halved crosswise, cut lengthwise into 1/4-inch-thick wedges

1 cup sugar

1/2 cup lime juice

One 750-milliliter bottle Riesling or other sweet white wine

1 cup gin, such as Hendrick's

1 cup triple sec, such as Cointreau

Two 12-ounce cans lemon-lime soda, such as Sprite

1/2 cucumber, halved crosswise, cut lengthwise into 1/4-inch-thick wedges