The President doesn't really need Congress to close down Guantanamo. He just has to do it. The money he requested, which congress denied, can be found from other sources. Just do it.
From the AP (via TPM):
Forty-eight terror suspects currently held at Guantanamo Bay are waiting to be released to other nations, the Obama administration said Thursday. The detainees are among 50 detainees whose cases President Barack Obama said Thursday have already been reviewed. The detainees would be the first to be released to other nations under the Obama administration's effort to empty the Cuba-based prison without bringing all its inmates to the United States.
Two other detainees have been released since January, to Britain and France, officials said.
Military officials described those detainees Thursday as either low-level threats who no longer have valuable intelligence to give, or have been cleared for transfer because of a court order or otherwise lacking evidence against them.
Whether they will be transferred anytime soon is anyone's guess.
The government has been negotiating with nations like Yemen and Saudi Arabia for months to deal with some of the detainees. Obama is seeking to place up to an estimated 100 Yemeni detainees in Saudi rehabilitation programs.