Use live paint. Fill the wanted shape with the live paint bucket. Then you can expand the object, ungroup and move the original ellipses and lines away. Here the bucket fill is just done. The red color shows what part is under filling:
Live paint is live because it's not final, it allows editing until it's expanded. Expanding generates a new shape which is the fill. Ungroup, before trying to move the fill or the original boundary curves. It can bee a good idea to save them a while, only shut them in the layers panel.
Here the object is expanded (not strokes nor the fill), ungrouped and mowed a little apart:
Your other option is to complete the lines to a polygon, join the lines and in pathfinder panel subtract
- the smaller ellipse from the bigger one
- the polygon from the difference of the ellipses.
Then add a normal fill.
A more worksome method is to split all lines and curves at the crossings and rejoin the wanted shape which can be filled.