For me, that sounds bad after wonder. I would never say "I wonder that..." but rather something like, "I wonder if/whether." There doesn't seem to be anything wrong with, "How did you wonder whether Bill fixed the car?" to me, but maybe other people won't agree. I'm not even sure myself. I saw this thread this morning and some times when I think about the sentences they sound moderately bad, sometimes awful and then sometimes passable. But it did occur to me that that basically never occurs next to wonder and I wondered whether or not that was the reason.