I've got a simpe request which perhaps even the remaining inept Republicans can pull off. Get Sotomayor to talk in public. Sure, it'd be nice if you could get her to reveal her reasoning on various legal issues, but I think a lot could be accomplished just by getting her to talk about anything.
The Democrats are so happily stuck in identity politics that they don't see anything else. To them she's a woman and a hispanic- therefore she must be brilliant. I doubt anyone can actually block her rise to the Supreme Court, but could someone at least sit back and chat with her for a few hours?
Extract some hint of the content of her character. No need to be aggressive- you could talk about the weather-just get her talking in those hearings about whatever it is you can get her to talk about. I suspect those actually interested in character rather than identity groupings will find it illuminating, if not downright embarrassing.