I think cycling should benefit your running. I think the problem is, you are not pacing yourself correctly for the goals that you have in mind. If your target is a 5k(3.1 miles), you shouldn't be going all out for 10 minutes every time you run. When I started to realize i'm not immortal in my early 20's and started running, I did the same thing you are speaking of. I went full gas for as long as I can and saw diminishing returns on my effort. Every time you go all out, you put stress on your legs and need recovery time. When you do that on consecutive days, you don't recover properly, end up being slower due to muscular fatigue, and put yourself at risk for injury. That was me and I gave myself a stress fracture that took months to heal.
In my opinion, you need to spend more time at a lower intensity to reach your goals. If you train the right way, cycling should be for recovery days and running should be what you're recovering from. I'm no star or expert in either, but decent at both. Here is my advice(do these in sequence):
-Make sure you have proper running shoes and bike fit. If you don't have the right equipment, you risk slowed improvement at best and injury at worst.
-Disregard this one if you won't be able to cycle much due to weather,temperature,etc...Don'd do any running and focus on cycling for a month or two. this will give you time to rest your legs,become more acclimated to the bike, and gain a good aerobic base with a low impact form of exercise.
-Run for more than 10-15 minutes at a lower intensity or as long as you can 2-3 days a week and a mix of sprints, walking(think HIIT?) once a week maximum. running 3 miles will become much easier for you with proper pacing and a higher aerobic base. Increase the duration when you feel up to it and make sure to stay hydrated.Cycle if you want to train beyond this and get 2 recovery days in, 1 at the least.
A few things:I personally hate hydration belts and packs, but make sure you bring something to drink while running. They have these hand straps for water bottles that I've used for hiking and running before. For the distance you will be going, a 10 oz to 20 oz handheld bottle should be fine. This summer is when I started cycling and ran maybe 3-4 times since then and was pleasantly surprised at the results. My first run in 3 months and I was able to do 4.5 miles at a faster than usual pace without destroying my legs or lungs. Cycling will give you muscles that you wouldn't really get otherwise(just by running) and really gives your legs a chance to recover.