Is it possible to remove Zubber from tile and carpet?

My 3-year-old got some of the colored portion (unmixed) of a Zubber kit on her shoe and tracked it all over our apartment. We have green polka-dots on the rug now but that isn't even what worries me. It's the kitchen tile. The color is gone, but a very thin, dry, slippery residue remains. I figured from the feel of the stuff that it must be oil-based but a scrub with dish soap failed. The floor still feels like non-stick cookware and a little research online has revealed the reason. The stuff is made of silicone! And according to still more research, it takes heavy duty chemical solvents to remove silicone.

Does anyone know something a little more kid-safe that might make the kitchen floor less dry-slick? It's like whole sections have been dusted with baby powder. The 3-year-old already fell on her heinie in there, and while that might sound like karma, I would prefer if it didn't happen to anyone else. The residue on her shoe might scuff off on the sidewalk, I hope. But I can't go rough up the tile.

I'm skeptical about my chances with this but I figured I'd better check around before going out to buy some big rugs.