Disaster Zone - suggestions needed

In the spring, if you'd have asked, I would have said our office space was a disaster zone: a mess of random papers and other junk that made the desk unusable (which was fine because no one needed to use it anyway). It wasn't for lack of trying. We have had a decent file cabinet and workable filing system for a while, plus we try to print and save as little as possible. But sentimental knick knacks, a cardboard box of baby photos from our parents, old cables and hard drives that aren't quite trash (or needs to be destroyed/wiped first), well, it takes up space:

With Herculean effort, we cleared the desk in anticipation of giving it to someone else who needed it more. We mostly found homes for everything (or stacked things in the corner) and gave some of it away (seriously how did we have at least 5 packages of lined paper?? do people even take hard copy notes anymore). And we did give the desk away. At least, we tried. It got halfway out the door and then gave up the ghost for good. It was a Craigslist Ikea find that had already moved from our old apartment, so one more move was apparently too much. So we moved it to the curb for trash pick-up then, instead of to its intended recipient.

So did our disaster zone improve? Not one bit. It just transformed. During the work on the breezeway (a room we have now taken to calling the "sun room" since it's beautifully sunny in there), we had to move all of the toddler's toys out of the living room. And so they migrated to the office, where they promptly become "out of sight, out of mind." It's quite nice having no toys (ok, who am I kidding, let's says *fewer* toys) in the living room. But now the office is a mess!

The small things that can be categorized are more or less "away." There's a train box, a puzzle box, a lego box, etc.

We could use perhaps a little more space for boxes like these, but the real problem is there are also tons of big toys. Toys that don't fit into cabinets. Multiple strolls around Ikea have failed to provide inspiration. Pinterest searches haven't helped. And so my question o you is how do we organize the toys so that this room is not a giant mess. We have one almost empty wall and would love to hide these toys in or behind some kind of furniture to pretend like grownups are still making the decorating decisions around here.

So put your ideas in a comment and help us out! And yes, while "throw the toys away/donate them/hide them in her bedroom" might seem like good options, they're not in the plans right now. So let's assume we are keeping them and just need a way to store them more efficiently in the office. Bonus points if whatever organizational system you suggest is also toddler accessible, to avoid having a constant stream of "mommy, can you get me my X?" "daddy, I want Y." and also lets us make *her* put them away.