The Judicial Branch of government is the only branch willing and able to to resolve this matter.
Will they do it as justices of the highest court in the land in an impartial manner or will they play politics with it?
They may be the only branch of government willing to tackle and resolve this issue, but they are not the only branch capable of doing so.
It only takes one member each from the Senate and the House to file a challenge to Obama's Natural Born Citizen qualifications for the Congress to initiate proceedings.
Will they do it as justices of the highest court in the land in an impartial manner or will they play politics with it?
The high court isn't given to playing politics with any issue. The individual Justices often have different interpretations about what the Constitution means, but they take their position as sole arbiters of that document very seriously.