It makes your life fifty-billion times easier if you abandon all efforts to store the lids with the box they go on.
New system - all plastic food containers live in a largish box on a shelf, and beside this box are two smaller boxes containing just lids.
You don't need fancy boxes - my 'lid' box is made out an old cereal packet cut in half - the lids slot in vertically (vertically is important in a Kondo stylee so that you can see what's there and remove without disrupting any other lids).
The containers live in a large, er, plastic container, but an average cardboard box would do. I don't attempt to sort them by size or anything, just make sure the box is roomy enough so you can sling the clean containers in with speed and impunity.
It's unbelievably easier - just thought I would share.