What is the fastest way to improve 2 mile run time?

It is mental. I would recommend that you approach the APFT with a strategy in place. That means finding the exact number you want to hit on push ups and sit ups that would not compromise your performance on the run. So, hit your number and get up if you can achieve a decent score in those events without putting too much stress on your body. To get better at the run to improve time, I normally developed a plan built around mastering a set time. For instance, I would take your time of 18:18 and run it after doing 20-30 push ups and 30-40 sit ups (with the normal rest periods like the APFT). The goal is to run without stopping for 18:18 with no regard to distance. You want to do this event 3 times a week (MWF) while the other days is mixed with an interval and long distance run. The first thing you do want to do is determine your pace, so make sure you track the first mile on your first run to get an idea where you are at (that is the only time you would track distance). After that, you want to check it again each Friday to check for improvement.

This is something that worked for me and there are plenty of ways to improve, but the bottom line is that you have to run after doing some other physical activity. You have to get use to running while being a bit stressed from other activity. You are trying to shave off 2-3 minutes in less than a month while improving in other events. It will be very important for you to go into the next test with a strategy.

I've always enjoyed my 60/120s and hill repeats. A friend of mine once told me, "people are strong but they're not strong when they're winded" and it's true. He had me do 4mph on my treadmill at 12% incline. minimum 4 miles. BUT I'd have to walk for 90 seconds, jump off and do 5 burpees in 30 seconds and jump back on. This was a 2 minute circuit and by the end of my first mile I hated life, by the end of my second I was pissed. By the end of my third I was seriously reconsidering my life choices, but I only had one mile left so I might as well keep going.

Doing this twice a week, 60/120's once a week, one long distance run, and one 2 mile run per week with two rest days a week for a month and I couldn't believe the difference it made and I got my fastest run time ever.

Wanted to post an update that following the advice here, I officially ran the 2 miles as part of the ACFT in 16:59!