There is a lot of training information out there. Some people say, "You don't have the speed! Work on that with these hills and sprints!", while others say, "You don't have the endurance, so increase your mileage and make your long run 10 miles!". In reality, a mix of all kinds of training methods plus dedication and consistency are what helps. I started running in 7th grade with 20mpw and a 24:40 5k. By 8th grade, I was running 35mpw and 5:18 1600, 11:50 3200 , and 18:30 5k. I have not run any more than 35mpw since. Now, as a sophomore, I run 4:37 1600, 10:13 3200, and 16:52 5k. I improved by 1.5min in the 5k without changing my training. I only grew 1 inch and 5lbs since 8th grade as well. I didn't gain much muscle. What helped was consistency and dedication.
I feel like what you're doing is looking for a quick fix to your apparent "slowness" and that it is a problem that needs to be fixed. Your current 5k time, 22:45, is not slow, it's just a step on your progression. As you said, you have only been doing 15 to 25mpw. I will now outline some steps that will make you a stronger and more dedicated runner, and then I will provide some guidelines for you to make your own specific training plan. This is a process which takes time, and luckily it's summer and you have plenty of that!
Step 1: Start a training log. Unless your browser messed up, if you go to the bottom of this page, there is a link to FINALSURGE, a running log that allows you to do all kinds of stuff, and its free! Logging your training is key for consistency because it keeps you regimented. You'll see "30mi last week" and you'll know that you have to hit 35 this week. You'll be able to track your progress as well.
Step 2: Start running! For the first 2-3 weeks, don't worry about a specific training plan, just get out the door 5 times a week. (or 6, but 5 is probably good for you now). Set loose mileage goals for each week, but don't do any tempos or fartleks or intervals or whatever just yet. Just run some miles and log them on your new training log.
Step 3: Start a specific plan. At this point, you should have worked your way up to about 25mpw, and now you can start specializing a bit. I have worked out a guideline for coming up with a schedule below:
Mon: easy run
Tue: tempo
Wed: easy run
Thu: tempo or fartlek
Fri: easy run
Sat: off
Sun: long run
--Easy runs are medium length runs at easy pace. Easy pace is the pace at which you feel like you could run forever. You should be able to hold a conversation with a friend while running at this pace. I estimate your easy pace at 10:00/mi or so, but you should go by feel.
--Tempo runs are medium length efforts at your lactate threshold pace. That is the pace you can hold for an hour-long max effort. For you, that is around 8:15-8:30/mi. A Tempo run is around 15-20min for our purposes.
--A fartlek is an easy run that has short, fast bursts in it. Do your normal easy run but every 2min or so, throw in a short (30sec ish) burst at mile pace. It's like an interval workout, but it is less tailored for a specific purpose other than general fitness.
--Your long run should be 25-30% of your total mileage. If you're running 30mpw, that week's long run should be 7.5-9mi. This is at easy pace. As you get more advanced (sub-20 5k level), you can make the long run into a progression run. Don't get ahead of yourself though.
Your mileage progression throughout the summer should look something like this: 15, 20, 25, 15, 25, 30, 35, 25, 35 and hold 35 until school. The down weeks are important. Give your body a rest every couple of weeks.
Those exercises you said were too easy? They are still good for you. If they are too easy, try increasing the number of times you do it or increase how long you do it. Stuff like bodyweight squats, calf raises, planks, push-ups, and light arm exercises are really good for injury prevention.
Lastly, to handle this increased stress on your body that comes with the extra running, you need to eat healthily and a lot. Forgive me if this is a stereotype, but a lot of the female runners I know interpret healthy as: "I'm going to eat fruits and vegetables and maybe some bread, but only before a race." That is wrong. You need a balanced diet, and a lot of it. I usually eat 3500+ calories a day if I can, and I only weigh 130lbs! Here are some tips for eating properly:
1. Don't shy away from carbs. They give you energy. Eat bread and pasta, and stick to whole wheat *if possible*. Beans and root vegetables are also a superb source of carbs. Don't get carbs from processed junk like sweets, etc. Healthy carbs should be 50-55% of your daily caloric intake.
2. Eat a lot of protein. Lean meats like chicken, turkey, and fish are your friend here. Also, beans (them again!) are really good, perhaps better. Eating a bit of red meat is also fine, as long as it's lean. You need iron from the red meat, especially since females lose some iron with menstruation. Cutting back on saturated fat in your protein-containing foods allows you to supplement with healthier fats. Protein should be 15% of your daily caloric intake.
3. Fat is not bad. Just make sure it's good fat. Getting fat from vegetables, nuts, and seeds (monounsaturated fat) is the best option. Dairy fat is a lot better than meat, but still not quite as good as from vegetables. Fat should be the remaining 30% of your daily calories. Having a healthy amount of good fats speeds recovery.
4. Dairy is very helpful. It's not the only way, but it is by far the best way to get protein, calcium and vitamin D into your system. Calcium and vitamin D will help with bone growth, which is good because it will fight stress fractures. Chocolate milk is the best recovery drink to have right after a run. You walk in the door, get water, then get some chocolate milk! Try to avoid whole milk if you can, instead drink 1% or 2%. Skim is not great since a tiny bit of fat helps absorb the vitamin D (a fat soluble vitamin).
5. As I said earlier, eat a lot. Don't be afraid of getting fat, since you're running. Don't weigh yourself to monitor fatness since you'll be gaining muscle. Instead, just watch your muscle definition. If you're getting a bit pudgy, bump your mileage up a bit and decrease your portion size just a tiny bit. Don't cut fat or carbs from your diet, just eat a little less overall.
Being a distance runner requires time and dedication year-round. It takes discipline. Based on your times off of an apparent lack of training, you have talent. Don't squander it because you don't want to put in the effort. Hopefully all of that is enough to get you started, but feel free to ask questions. I love helping people with running. Good luck!
tl;dr: I spent an hour writing this and making sure it's good when I should be studying for my history exam. Humor me and read the whole thing and consider it seriously.