I'm looking for some advice on whether to apply for an LLM or J.D. after completing my LLB.

I'm a US citizen studying law at the London School of Economics and on track(fingers crossed) for a first class degree. I want to move home to NYC. I know that I am eligible for the Bar exam in NY with only my LLB but I definitely want to augment my UK degree with some American education.

Will the LLM be sufficient for my purposes, or should I apply for the JD? I have read the other posts about LLM vs. JD decisions but most of the answers seemed to pertain only to non-US citizens who are not offered jobs after the LLM because of the obvious problems associated with visas, etc.

Any help would be much appreciated!