Modifying a curve using point handles

One way to modify the curve of a vector shape is to use the point handles. These handles control the direction and amount a line curves.

Click the Subselection tool (the white cursor arrow you are currently using) on the left-hand side curve point of the ellipse.
The point handles will become visible. (The small blue lines with circles on their ends)
Click and drag the lower point handle (the circle at the end of the line), until the lower curve is just below the corner of the rounded rectangle. If you move the point handle down to the horizontal guide and keep it roughly vertical you shoud manage this.

As you move the point handles the blue outline gives you a preview of where the edge of your shape will be.
Do the same to the right-hand side of the ellipse.

Don't worry too much if they aren't perfectly symetrical. If you had time you could set up some guides at specific distances to help you but for now we'll just do it by eye.
Choose Select > Deselect.
Choose View > Guides > Show Guides (toggling off the guides)