You were on the right track until this point:
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0.1 x 98 = 9.8 (The weight of the remaining water)
9.8 + 0.2 = 1 (The total weight of water and flesh)
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It was a bit redundant to multiply 0.1 by 98, then add 0.2. You could have just multiplied 0.1 by a hundred. But the point where you really went wrong was in saying that 9.8 + 0.2 = 1, instead of 10.
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Well, to begin with there is a total weight of 20kg - 200g of "dry melon" and 99 times that of water, or 19,800g. If after a week the water is down to 98%, the percentage that is dry melon must have risen to 2%. This means there is now 49 times as much water as melon, 9800g, and the total weight is 10kg.
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Ench wins the prize!