I have some additional things to add to this idea as well as I am doing something that is a necessary component of one of my jobs that interferes with my strength gains. I am a group fitness instructor at 24 Hour Fitness out here where I live, and I teach two times a week a class called "Body Pump", which is a endurance based lifting program choreographed to music. I have always done this for almost 8 years, and as much as I want to be as strong as I possibly can be, these 2 classes a week are also very rewarding for me to teach.

My solution is to make sure I lift at a level in class as in instructor that I do not get fatigued from teaching - for years I didn't do this. I do not want to expend my limited resources and allocate them into being a subject matter expert at my endurance based Body Pump training. I enjoy teaching the program, but I definitely have to make sure I am not expending too much energy into it. I also used to try to balance running, Body Pump, and strength gains into one regiment. Terrible idea. Not enough allocation towards recovery.

For the original poster, I can also say, since I have a very decent track background, those strength gains will carry over very strongly into the 1 mile run. Keep up your strength focus! I'm curious how your mile ends up doing.