I've got my LLM and am in the middle of my JSD at CCN. The LLM's a one year degree. Outside of tax, it can be useful if you practice for a few years and then decide to make the jump to academia, but only where your JD isn't top 3/6, maybe 14. HLS has an excellent LLM for US wannabe academics; so does Yale. Stanford is excellent, of course, particularly if you're interested in high-tech stuff. Chicago's great if you want law & econ, antitrust, regulation etc. Col & NYU have fine programs too.
The JSD probably isn't particularly appropriate for US JD graduates. The JSD typically takes 3-4 years to get AFTER completing the LLM degree. Of course, having a JSD (Ph.D in law) is helpful in seeking academic positions, but only because it tells law schools you have the ability, aptitude and work ethic to engage in sophisticated legal research. The truth is that, as a JD graduate, you'd be far better off concentrating on putting together a record of publication, which, at the end of the day, is what law schools care about in making their hiring decisions.
I plan to go into private practice for 3-5 years, after which time I'll probably look at the academic market. I'm not a huge fan of academic writing, but I know I'd love teaching infinitely more than I would billing 60 hours a week on a regular basis in a big firm.
Let me know if you have any further qs about LLM/JSD programs.