It seems like you're poor at cardiovascular exercise/aerobic movements but aren't unfit in the broader sense.
You've got an elevated bodyfat level, which means your lean body mass (LBM) is low. To fix this, you'd want to start training with weights. Compound exercises to get your LBM higher, you'd also want to do quite a bit of cardiovascular exercise if you're looking to run 13 miles. If you can't run half a mile yet, you've got a long way to go.
I'd recommend a 3 day routine, with an off day in between. You can use Saturday and Sunday as cardiovascular days in which you run/brisk walk/swm/cycle etc.
Here's a breakdown of each workout
Weights:
3 sets x 5 repititions.
First of all, don't worry about getting bulky. You won't. Especially if you're a girl since you don't produce enough testosterone. People work very hard to get bulky. It won't happen unless you try really hard.
Weight should be adjusted to a level which you can only lift around 6-7 times. Weight should be increased every workout. Yes, that means every alternate day you increase the weight by an amount. No exceptions. Generally it's 5lbs on the Bench Press, 10lbs on the Squat and 15lbs on the deadlift.. The rest vary by a lot so you should play around with them yourself.
Workout A: Squat, Bench press, deadlift (Deadlift is an exception to the rule, should be 1 set and 5 repititions.)
Workout B: Squat, Overhead Press, Bent over Rows.
Cardiovascular Exercise:
Since you're planning to run for a long distance, that's what you should do. Training for endurance involves training for a maximum amount of time. You'll gradually see that the 'maximum' will keep pushing itself forward the more you train. The amount you train will therefore increase, and you'll be able to run the 13 miles.
I'd dedicate an hour or two to cardio on weekends. This could be a combination of many forms of cardio or just running. If you want to run, running would be the best option. Don't stop, even if you feel like you're going to collapse. It'll help you break the barriers and move on to the next level.
Diet:
Vegetables, fruit, milk, meats, whole grain bread, oatmeal, etc.
Good luck.