Here the problem is replicated and solved:
At left there's 2 filled open shapes. In the middle the fill was removed from the open shapes. At right everything was selected and a Live Paint Pucket was applied to the white space. The red lines are markers showing which area will be filled.
Live paint is tricky because it is not final. The shape must be expanded to fix the effect. Do not expand fills and strokes, only the objects. That makes the painted area to a shape. It's grouped with the open curves. Ungrouping makes possible to delete the open curves and leave only the filled closed shapes.
Another way is to duplicate the closed circles (=select, copy, paste in place) and with the scissors tool cut the circles at the crossing points with the open curves. Then delete the extras of the duplicates and join all open curves to closed paths.
In the following image a grey stroke was added to show what is joined. You can join 2 curves at a time.
If you want to get rid og intermediate black strokes and have a solid single shape, goto the pathfinder panel and unite the shapes.
The following image shows what is achieved