Litchfield Park is a popular day trip from Darwin with a bit of most of the local terrains and wildlife, except salties 'cause there are a number of swimming holes there they do try to keep them out. Further down the road is Edith Falls near to Katherine which is good for birds and swimming. But you would be mid-wet season which may make access to some places difficult and the great outdoors hot and sweaty. And swimming can be risky at that time, too easy for salties to move around without being spotted. Specatcular for the waterfalls, though, if you can get near them. Probably not enough time to get to Kakadu and back and actually see anything worth seeing unless you enjoy nothing so much and driving and jogging in tropical heat. But Howard Springs and Fogg Dam are in that direction but closer to Darwin. Personally, I find Darwin itself fairly easy on the eyes for a small city at least along the waterfront. There is (or at least was) a bit of a regular fish feeding down by one of the boat ramps. Is it Fannie Bay? Can't recall. Nice for a sunset and mangroves not far. Not sure if the jumping crocs tour will be running, but that would be an easy way to see a bit of the area and maybe get an interesting photo if they are. I really hope you have a great time.