Not having a quant background really won't hurt you, or at least that what's I've been told my many profs--especially that the programs to which I have been accepted. Most profs won't go out and say it explicitly, but a JD from a top school definitely says something. Your publications may have an impact, depending on what you want to study. I think polisci is more quant focused, and programs DON'T want you write and research the same way you did in law school.
I don't really know how grades will factor in, but if you go to HYS or even CCN, you probably don't have much to worry about, so long as you have above a 3.0. Like I said, though, that is a guess.
Your letters will be fairly important--you want them to be as good as possible. Pick wisely.
The most important element of your app is your SOP--BY FAR. Concentrate on this, rather than factors over which you have no control. A good SOP goes a long way.