Eisenberg is right. That will indeed help by strengthening the muscles and tendons that create the arch of the foot. It won't always get rid of your flat feet; your arch may still collapse even if you have strengthened the support muscles. But it will always be beneficial to your health to try, though. You have to learn good form first, though. Just beware; your calves will be sore as hell in the beginning.
running on the balls of your feet is what you're supposed to do, but some people who are WAY flat footed like me have no choice 
This is where the shenanigans-calling belongs; there's no way that you are so flat footed that you can't run on the balls of your feet. Your arch would have to bend backwards past 180 degrees to make it possible. It's okay if your heel touches the ground, it just has to touch after the ball of your foot. The most important thing is that your foot contacts the ground right under your center of balance, not in front of it.
I suggest you try learning good form again.