You could maybe get some stylish storage bins. I saw some at westelm.com (just do a search for storage bins). If those are too pricey, or don't complement your decor, you could probably make some (could be a fun activity for the kids!).

Can you put doors on this shelving? that way you can hide the contents when you need to. you will need to organize contents and store them by frequently used or if you child reach area. The toughest part is always trainning the uses to put things back where they got them from. I like roll out type storage so you can get to the back of the storage more easliy and storing thing verically like a file cabinet so you can see all the items. City Mill, Home Depot and even Price Busters has great storage containers and storage ideas.

Maybe you could use some decorative storage bins for the kids things. I always find nice ones at Ross's.

Try to organize things so that you can efficiently use every bit of square inch of that entire pace. This means getting organizers, boxes of different shapes and sizes that would fit together like a puzzle. Label the boxes too so you can easily find things and not waste time searching through an assortment of things. City Mill, Ross' and even Costco have good organizers.

Looks like you don't really use this as a desk... more of a storage puka. Would it be possible to just remove the desk and what ever is above it adding a bi-fold door that opens to the left (looks like a stove is to the right) and just make top to bottom shelves more like a pantry closet. That would add way more storage space and you wouldn't have to see it at all.

I have the same in one area of my kitchen. I put a plastic storage bin under the counter, removing the wheels to fit, and would suggest a nice fabric curtain. If this does not look right, I like the idea of shelves with small areas to fit what you have, and add doors to match the other cabinets.

What a about a cabinet with doors, or a divider ?

I agree with mauigirl. Storage is what you need. Take out the desk and put shelving from top to bottom. If you want to be organized, then get storage bins for each child.