Remember when you ran to catch the school bus as a kid? The chase only lasted a few seconds, but you were still winded. That's the type of exercise almost nobody does as an adult, but if you want to improve your sprint time, you need to work on short bursts of explosive exercise.
You can develop your body's energy system responsible for this sort of explosiveness with 4- to 6-second drills, and it's not necessary to perform sprints at first. Instead, kick it into high gear for 5 seconds on a bike or move your feet back and forth over a line as quickly as you can for 5 seconds will work. Watch this video as an example.
These drills don't require you to decelerate your body like sprinting does so you're less likely to blow a hammy, but you'll still condition your mind and body for speed.
Work on mobility, stability and strength, which all contribute to speed as well, and then try progressing to short sprints and work on your running technique to shave seconds off your time.