Also, while I have run considerably faster in the past, I am in roughly the shape that you set as your goal. I have run 3 5ks since August in the heat and humidity of the southeast. All were between 16:07 and 16:09. Here is a smattering of my workouts from that time. Not in exact order, and more speed oriented than I would recommend for you (I stuck to shorter workouts because it was so hot and humid where I live).
6 x 800 at 2:30, 2:29, 2:28, 2:29, 2:30, 2:30 w/2:00 rest
4 x 1600 @ 5:11, 5:17, 5:17, 5:19 w/:60 rest (cruise intervals - tempo workout)
4 x 1200 @ 3:43.7 avg w/3:00 rest
2 x 200 (31.9, 35.2) w/1:00 jog; 2 x 400 (:66.7, :66.7) 2:10 jog; 600 (1:44.1) w/4:30 rest; 600 (1:45.1) w/2:30 rest; 2 x 400 (:70.7, :69.1) w/2:30 rest; 2 x 200 (:31.1, :32.1) w/1:00 rest
2 up; 2 x [4 x 400] with 2:00 rest and 4:00 between sets (67.8, 69.6, 68.3, 67.2)(67.9, 69.5, 68.7, 67.1); 1 down
16:08 5k
5 x 1000 w/2:30-2:55 rest all at 3:07
2 up; 4 x [200 (:33.1-:34.4), 200 slow jog, 800 (2:29.4-2:30.6), 200 slow jog, 200 (:34.5-:35.6); 1 down
2 up; 2 x [4 x 400] with 2:00 rest and 4:00 between sets (67.8, 69.6, 68.3, 67.2)(67.9, 69.5, 68.7, 67.1); 1 down
16:08 5k
1.5 up; 8 x .25 (urban trail) @ :69.6, :68.2, :67.9, :68.3, :68.9, :69.9, :68.0, 67.9; 1.25 down
1.5 miles up; 3 x 2400 cruise intervals (8:04.1; 8:03.9; 8:08.0); 2 miles down
2.25 up; 5 x 1000 (3:06.1, 3:08.2, 3:06.6, 3:08.9, 3:10.4) w/3:00 mostly active rest; 2 down
20 minute tempo run (5:23)