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Here's a pipe-dream, but perhaps the 20th Amendment has wiggle room for replacing an administration by means other than succession?

Section 1. The terms of the President and Vice President shall end at noon on the 20th day of January, and the terms of Senators and Representatives at noon on the 3d day of January, of the years in which such terms would have ended if this article had not been ratified; and the terms of their successors shall then begin.
Section 2. The Congress shall assemble at least once in every year, and such meeting shall begin at noon on the 3d day of January, unless they shall by law appoint a different day.
Section 3. If, at the time fixed for the beginning of the term of the President, the President elect shall have died, the Vice President elect shall become President. If a President shall not have been chosen before the time fixed for the beginning of his term, or if the President elect shall have failed to qualify, then the Vice President elect shall act as President until a President shall have qualified; and the Congress may by law provide for the case wherein neither a President elect nor a Vice President elect shall have qualified, declaring who shall then act as President, or the manner in which one who is to act shall be selected, and such person shall act accordingly until a President or Vice President shall have qualified.
Section 4. The Congress may by law provide for the case of the death of any of the persons from whom the House of Representatives may choose a President whenever the right of choice shall have devolved upon them, and for the case of the death of any of the persons from whom the Senate may choose a Vice President whenever the right of choice shall have devolved upon them.
Section 5. Sections 1 and 2 shall take effect on the 15th day of October following the ratification of this article.
Section 6. This article shall be inoperative unless it shall have been ratified as an amendment to the Constitution by the legislatures of three-fourths of the several States within seven years from the date of its submission.

What if Congress were to find that the IRS scandal was so vast that it had clear material effect on Romney's rights? What if they were to further find that the IRS was also misused to coerce Roberts' vote on Obamacare?

If Obama and Biden were both to be impeached and removed, couldn't Congress decide that the better approach going forward is to declare that Obama/Biden failed to qualify in hindsight, but that their fraud was only exposed now? Therefore, finding that they never qualified in the first place, Congess will exercise its 25th amendment power to declare that the other candidate (who, let's not forget, won 50 million votes) is now president.

Sure, the Senate would balk at the idea. But they can't just waive away 50 million votes as if it had no bearing on the matter.

-PJ