What can I do about mild water/moisture damage to laminate wooden flooring?
The most common approach to dealing with flooded surfaces is to blow air across them. The constantly changing air absorbs moisture. Commercial companies use very high volume fans, but any fan should help. If you can exhaust the air from the room, the moisture level in the air should drop and more water will be absorbed from the floor. 

Use of a dehumidifier would also speed up the drying.

While it is temping to use heat, there is a risk that the heat or the steamy water it generates may compromise the finish on the floor.