Exactly! Finally, someone sees what I see...
How perfect would it be for Mystique to be the one behind the Watergate break-in? Or exactly what you said, that she infiltrates Nixon's cabinet and gets information about this X Envelope that is locked up in the DNC Headquarters.
And the X Envelope could be anything for the purpose of this movie. It could be plans to build the sentinals, or Mastermold, or the inhibitor collar, or a list of all known mutants and their powers, or whatever Singer wanted it to be to make it a pivotal plot point.
If you want to read about this X Envelope, which again was the actual cause of the Watergate scandal, check it out here: