The consul has taken refuge in India's Permanent Mission, and I don't think the State Dept. is "desperately" looking for a way to free the consul from the legal fix she's in.
There's something most people don't seem to grasp here. Let me put it bluntly:
The chief prosecutor of the Southern District of New York outranks the State Department!
The Southern District of New York is special. It's the richest district in America's densest concentration of big businesses. This prosecutor is usually at the top of his profession and very well-connected academically, legally, and politically: that is, he recommends bright young lawyers to governors, presidents, etc. to serve as prosecutors, public counsels, etc. and experienced ones to serve as attorney generals, judges, even justices of the supreme court. It can be a powerful network. Such men have at times in the past moved on from CPSD to serve in the Cabinet or as Supreme Court Justices themselves. (ex: Cordell Hull, Felix Frankfurter)
U.S. law and international treaty gives the State Dept. very little leverage in this case. You think it might, because there are provisions in the Consular Convention that allow the receiving country some leeway in permitting "extra privileges". But in the U.S. most of that leeway is now determined by law and not subject to the whims of State Dept. officials.
So it's probably a waste of time for India to negotiate with the State Dept. since there doesn't appear to be anything State can do. India - or Khobragade, if India does the right thing and cuts her loose - will have to cut a deal with the prosecutor.
A country that launches its own interplanetary probes, fields nuclear-tipped missiles, wields aircraft carriers, and has its own powerful multinationals should still be considered a "developing country"?