I wouldn't necessarily call myself an expert on distance running, but I had some success in high school (4:39 mile was my best time). In college, I played basketball, and I was still able to run a 5:10 mile while focusing significantly more on lifting. Like others have said, while distance running and lifting don't always go hand-in-hand, you should be able to make strides in both at the same time if you really put in the work.
I'd recommend at least one tempo run greater than two miles each week. The longer you can run, the more comfortable you'll feel with shorter distances. So if you're currently around 13:30-14 for two miles, I'd try to run three miles at a slightly slower pace--maybe something around 23-24 minutes. While I don't think it's necessary to run 4 or 5 miles with your particular goal, you could do the occasional longer, slower run, as well.
In terms of workouts on the track, I'd stick with distances between 200 and 800 meters. Short sprints are great, but you need to be able to run at a pretty decent pace for a relatively long time to meet your goal. I might try something like 6x400m at your mile pace--probably something around 6 minutes, or 90 seconds each lap--with two minutes of rest between runs.
Other good track workouts:
3x600m at current mile pace with three minutes rest
6x300m at goal mile pace (slightly faster) with one minute rest
3x800m at goal two mile pace with two-three minutes rest
12x200m at goal two mile pace with 30 seconds rest
I hope this helps. Let me know if you have any questions.