ask an Federal Court to issue a Writ of Mandamus ordering the Senate to do its Constitutionally mandated job.

Article II, section 2, clause 2 of the United States Constitution empowers the President of the United States to appoint Justices to the Supreme Court, and requires the United States Senate to advise and consent in this process.
This petition asks the Obama Administration to sue the United States Senate in a Federal Court, and seek a Writ of Mandamus, ordering the Senate to do its constitutionally mandated job. This action would be appropriate, as a Writ of Mandamus is an order from a court, ordering a government official or body to do their required job. Because Article II, section 2, clause 2 of the US Constitution requires the President to appoint replacements to the Supreme Court, with the advice and consent of the Senate, the Senates refusal to do its job is a violation of the law.