Firstly...this is just my opinion..so please don't kill me if you disagree: You should reconsider the law school decision. With your LSAT score and GPA you should reply to the top schools and see where the chips fall. I dont think that there is any policy arena in which a degree from HLS or Columbia Law is less valuable than HKS or WWS.
In regards to your question: I got into WWS and HKS. We have similar verbal scores, but my quant was significantly higher. I came from a polisci background so I put in some effort to score above 160 for the quant. My understanding is that the median for the top schools(HKS, WWS, Goldman) is around 161Q and 162V. But I also had a very quantitative job, so I think that was a factor.
Honestly from your profile, I doubt that a few quant points or classes made the difference in your app. Sometimes you are just not a fit for the class that adcom is trying to build.
I am grasping at straws, because I don't really know what to tell you. You have great scores and you seem like the perfect candidate. I honestly think that this process can just be very random..maybe the person that read your app was just having a bad day..maybe there were too many education policy apps...what I am trying to say is that this occurrence, as sad as it is, in no way reflects your abilities or capabilities. My final suggestion would be this...try to highlight your interest in education through work experience and get your quant to over 160. And perhaps reconsider law school...especially with that LSAT score.