Illustrator :: Breaking Large Shape Into Several Smaller Shapes?

I have this object (shape path) that is like a continous loop or ribbon. see below:
 
I would like to break it into two seperate shapes: a top half, and bottom half. I need to remove some between them though.
 
Unfortunately, I cannot make the scissor tool, knife tool or cut path tool work correctly. It just doesn't seem to work, no matter whether I have anchor points selected or not.
 
I have also tried duplicating the object and trying to delete anchor points until I'm left with one of the shapes. However, because of the ordering of anchor points, I end up with a blue line connecting one end of the object to the other.