Hi there, Dr. 90210!
I was in a similar situation at some point - I ran track in college, then quit and gained quite a bit of weight - went to 140 pounds from 114 (I'm 5'5''). It wasn't just because of not doing LDR anymore - it was mostly due to my poor diet choices. I lost that weight now and have been maintaining it for a while.
You don't say how old you are, and that would play into how many calories you should it per day. You can try calculating your BMR and activity levels to determine the optimal calorie intake, but 1,200 does sound like a reasonably low number for the activity level you're planning on sticking to.
So (those are all rough estimates!), running 5K 5 times a week will result in something like 500 calories burnt per run (I'm assuming you run at 6 mph, but I obviously don't know how fast you run). If you eat 1,200 calories per day and run that much, plus add strength training to that, you should have no problem losing 1 pound per week. It does depend on your BMR, but if it's not much lower than 1,200 calories, you basically will be eating just enough calories to maintain your critical body functions, and then losing additional calories through exercise.
What you are planning on is similar to what I did, and it took me 7 months to lose 25 pounds. I restricted my calorie intake and increased my aerobic exercise intensity and frequency. It took a while, but worked really well for me - I stick to my calorie intake limits and keep exercising, and finally it looks like I'm not getting my weight back.
I'm really interested in what your progress is going to be like, hope you'll be willing to share!