I had the same thing happen when I made fermented beets (not beet kvass). It was like a layer of stringy goo with some mold on top, but I just scooped it off and we are still eating the beets -they're fine! I too have become much less squeemish about some foods since using the book!
We need to boil the water we use. I live in a town that does not fouridate or chlorinate. I still boil the water I use for any laco fermenting. Maybe using too much culture? Hope you figure it out. I've never experienced this.