Maybe you don't need to run every day? 3-5 runs a week should be fine for training, with park run as your long run, and a speed session in the week? Maybe follow one of the 5k training plans on this site, since that will give you an idea of the kinds of sessions you could do. Perhaps you could do strength training on your non-running days, since I'm sure there are other elements to the army fitness test. Good luck!!
You weren't far off, but you were expecting a lot on relatively little training. You knocked over 3 mins off your 10K PB in 12 weeks, which is really good, just have another go, but maybe work up to some longer runs 3-5 weeks out from the race. Make sure you can run 14-15 miles. HM isn't like a marathon, you can go longer distances in training without risking injury or burn out, but it reduces the risk of the wheels falling off on race day.