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Here's what I learned after the great mildew epidemic that followed Hurricane Katrina…
An electric dehumidifier will help dry the air and help keep the mildew from growing any more. A heater in conjunction with the dehumidifier will also help, by keeping the air temperature above the dew point.
Kitty litter can help absorb any standing water. A tray of kitty litter placed below the vent will also absorb future leaks until you can get the vent fixed.
White wine vinegar will kill the mildew on most surfaces. But if mildew has penetrated deep into cushions or mattresses, you're better off throwing them away; otherwise you'll never know if you got all the mildew from inside the cushions.
You can use an icepick to probe any damp wood to see if it has begun to rot. If it has, you'll have no choice but to replace the wood; rotted wood is permanently weakened.
Engineering: Finding complex solutions to simple problems you didn't even know you had.