As to Obama's recent recess appointments, the Constitution's Art. II, Section 2, paragraph 3, provides that "The President shall have Power to fill up all Vacancies that may happen during the Recess of the Senate . . . " (My emphasis.)
As to whether what Obama has done is a "Political Question" that a court would decline to review for jurisdictional or prudential reasons, I think it's not (see Powell v. McCormack).
If it's not, then there's a factual question whether there was a "recess." We Originalists probably wouldn't have much difficulty answering that question.
If there was not, the question then becomes one of judicial enforcement. But by then, there should be a new President.
So I see the entire imbroglio as moot.