How do I calculate how from from the "exact geometric center" one focci is in an ellipse?

Eccentricity is equal to the distance between the foci divided by the length of the major axis. With what you have, you can then find the distance from focus to focus. The foci lie on the major axis and are symmetric about the center. So your distance from focus to focus divided by 2 will give you the distance from focus to center.