"Evenly distributed" almost sounds like an aesthetical criterion. But I guess this could be achieved by picking points randomly. Say, you pick a point within the ellipse and attribute it as a center for a circle (take the biggest circle such that it still is within the ellipse). Then repeat the procedure, but now, taking into account that the new circle must not overlap with circles that are already present. Whenever no circle satisfies the criteria and you are not done yet with distributing all the circles, pick a new random point.
I guess the method is not very efficient if you're stuck with a big circle at the end of the day and all other circles have been distributed. One way to overcome that is to start with the biggest circle, determine the ellipse of points within which the center of that bigger circle can be placed and choose a random point from that. Then, pick the next biggest circle and go over the same procedure: determining an ellipse of points that can be possible centers and pick a random point again, but now you have to see that there is no overlap with the already placed circle. Etc...