If you're training for a distance race (5k or more) and not a sprint, then the best thing is to run a lot of kilometers. And the best way to do that without injury is to run slow. So it may sound strange but you learn to race fast by running slow, but running a lot.
Fitness
Do you want to increase your burst potential over a shirt period, say 30s-1m, or your potential for longer stretches, say 30m-1h?
I was thinking of a short run for a short race. I really want to impress my roommate the next time I see him. I could run for longer, but that's for another time when I start being serious about running marathons.
Not a fitness expert by any means, just an enjoyer. But from what you are telling here you need to improve your explosiveness/power. That will make you jumpstart faster to your highest speed. Maintaining it is another issue, but could be done by many means.
To improve your power, I would go with what another user suggested. Do some kettlebell exercises. You can also do some thrusters with dumbbells, or power cleans.
To maintain your speed, I don't know. I'm not an expert, least in running. But I noticed that the more I lift (in terms of raw weight) the easier is for me to run.
Of course, this is disregarding technique, which is key. You can have the best legs in the world, that without proper technique you'll get so far. Conversely, if you have the proper technique you don't need to improve your fitness THAT much.
Is a short race 200m, 400m or 800m+?
It might be worth it to check out the NerdFitness "how to become The Flash" superspeed routine here. It's got some good science behind its sprinting ideals (and how it also helps create endurance for long distance running too). It also provides information on how to train your speed up correctly so you don't end up injured or overexerted to the point of needing too much time to recover.
If you're mostly sticking to the treadmill, I used to do a variation of the 60/120s on treadmills to get my 2-mile time down. I'd warm up for a few minutes then jog for 5 minutes, crank up the speed and sprint for one minute, bring it back down to a jog for 5, crank back up for another one minute sprint, etc. It took longer than 60/120s but it was safer to increase the 120s to five minutes on the treadmill. (If that's something you try, make sure to find what's most comfortable for you to prioritize safety, as reaching the controls to turn down the speed when you start getting worn out and sweaty can get slippery.)
Nowadays I mostly work out indoors in a home gym setup. Because of that, I rely a lot on a combination of plyometrics, cardio routines, and martial art drills to increase endurance, breath control, etc. Try a few things, take what you like and mix it up, maybe substitute one of your running sessions with plyometrics or shadow boxing, find what keeps you interested and keeps you pushing in a way that brings progress.
Have you heard of the 'Kettlebell Effect'? Its where your performance in other sports is noticeably enhanced because of kettlebell exercises, specifically Kettlebell Swings. They train your posterior chain and give you strength, stability and explosive power all around the hip region.
I do strength training in a gym with machines, barbells and dumbbells, but ive also started incorporating doublehanded kettlebell swings to my routine. 20 swings with a 20kg bell and im gasping for air!
Getting my leg presses up to 8 plates per side shot up my running speed without any running at all. So I would say focus on leg strength and practice running a lot and keep trying to improve your form using videos or something.