In addition to the the other answers, also check to make sure your ice maker is not leaking around the the equipment box. Except for water on the floor, I had the same problem as you did, ice in the bottom of the freezer. It turned out to be the ice maker leaking, which is a common problem with that particular ice maker. I bought one on line and replaced it myself. If that is part of your problem, I can give you instructions on replacing it. It's not too hard.
I had the same problem. The ice maker started leaking water into the ice storage bin. I had to replace the ice maker. I did it myself and it’s easy. I ordered one over the internet. Here’s the link to the Sears parts: http://www.searspartsdirect.com/Kenmore-elite-Refrigerator-Parts/61005508/0046/464/Model-59676603600/0583/0161000.html
You’ll see three adjusting screws on the side of new ice. They preset at the factory so don’t touch them!
Ice maker replacement Instructions for bottom freezers: First, use your refrigerator to store the frozen food because it will only take you about 30 minutes to replace the ice maker. Unplug the power. Lift the ice maker wire switch to shut it off. One step no one tells you about but makes the job much easier is to remove the freezer sliding door. Remove the wire mesh storage tray in the freezer. Then with the drawer door pulled out, lift it up slightly with the handle, which releases it, and slide it out of the rails. Now you got it made! Carefully pull the front plastic cover off the ice maker. When you pull on it, you will see it separate from the rest of the machine. It simply snaps off and on the maker. You may need it for you new machine. Mine did not come with one. Unplug the wiring harness from the top back of the freezer. There are two hex head screws that hold the ice make to the side of the freezer. Unscrew them and the ice maker is free to remove. If your new maker did not come with the wire switch, simple pull the old one out of each side the switch mechnisiam and slide it into the new one. Now reverse the above steps to install your new maker. It may take an hour or so for it to start operating, just be patient. You may want to just sleep on it?