My suggestion?
1- Sort the the toys with the kids as angelshepherd said.
or
2- Take all the toys and put them in a large storage bin or two. Leave a few of your kids favorites. For a few days let them play with those and make sure they get put away every night. Then over the next week give them a few toys back and then a few more. Don't give them more though until they know how to organize the last bunch. So if the toys aren't put away no more 'new' toys next week.

(I remember as a kid my mom took some of my toys and put them in a box for about a year or more. When I got the box back it was like Christmas!)

May I make a suggestion? I have 4 children of my own. I had toys coming out my eyeballs. We really needed to get organised. It happend quite by accident really, but to make a long story short we seperated out things like match box cars and they got their own bin so did barbies, wooden trainset pieces and polly pocket type stuff. Then the kids were each given a small to medium sized tub for a toy box. We scattered every toy they owned out across the floor. And they were asked to fill their tubs with their favorite toys. The lids needed to fit on too. After that I got to go through and pick out a few things that were sentimental toys or ones i thought they may have missed. But it was tough, laying in front of me waere hundreds of dollars worth of toys that I had watched family and friends buy for my children. I knew deep down that they were rareley played with. I also remembered that all these toys no matter how sentimental, were the cause of many arguments with my children, many extra hours spent cleaning, many days of being over whelmed, many times of being embarresed by the mess, and just an over all sence of feeling bogged down. So after I carefully chose those few special items the children had missed I called in my backup. I knew the next step would be to hard for me to bear. The children and I went and had a coke while my husband eagerly discarded (to goodwill of course) all those toys. But when we got home the house had such an uplifted feeling. You really could feel the weight lifted from the room. And really the kids didn't even seem to notice. So from my perspective no matter how you choose to organise be sure to reduce first.

I discovered that building a simple storage system (from any of the several instructables) and having my girls help, made them more apt to use the system. They helped glue, hammer, and paint the storage boxes and shelves. We even labeled the different compartments. I was going to have them label the bins by the type of toys that would be put in them (i.e. cars, dolls, guns, etc.). I decided to let them decide how to label the bins. They went with colors. lol. Hey, it made sense to them. lol. Now the hard part, actually staying on top of my girls and helping them put away their toys as soon as they finished using them. I ended up temporarily putting the storage system in the living room so I could keep a better watch on them. We sang the clean-up song and practiced putting away toys. I even made a contest out of it, having them identify colors faster. They were rewarded with quarters that they could save to buy DVD's. The storage system is in their room and they put everything away. You'll have to check the bins every once in a while as I've found shoes, tools, and my wedding ring. lol. All in all it turned out to be a two part process. Providing a suitable storage system and building the "clean-up" habit.

Hope this helps

Good luck. I would suggest large plastic bins/boxes (labeled & color-coded) on sturdy shelves (secured to the wall to prevent accidents). It is hard to put things away when you run out of places to put them, and I can see that all of the bins you have are already full. When we were really broke I used cardboard boxes to store toys & games. They were easy to decorate with spray paint, in addition to being free.
The kids may eventually get more organized with storage space and bins at their disposal, or you may have to resign yourself to 20 years of chaos. Some personalities are happiest with organization, and some are happiest with everything in one giant pile (eeaaaghhh! how can you LIVE like this?!?! Are those truck parts in your laundry basket?) and a stack of empty organizers in the corner. Hopefully you did not accidentally name a child after a hurricane that was so devastating that the weather service retired that name :-)