But there was never a chance, with the Congresses Obama was handed with, that the prison would be completely shut down.You've added a critical word there with "completely".
Congress blocked Obama from bringing Guantanamo inmates to the US for imprisonment or trial. But 86 of the 166 inmates are never going to be tried. They're already cleared for immediate release. Everyone (CIA, DoD) has agreed to this. Obama has the power to release them with the stroke of a pen, and the State Dept. apparently has a plan to do it and is just waiting for the go-ahead. So while Obama may not have the power to close Guantanamo completely, he does have the power to free more than half of the prisoners there, the ones who are not accused of anything. Yet there they stay, for no discernable (other than political) reasons.
Pinning the blame on Congress has worked for Obama politically, but if it's true that he's the one keeping more than half of the prisoners there, and with weaker justification since they are the ones unanimously agreed to be innocent, then it seems fair to say that Obama is more at fault than Congress is.
Source: http://ianmasters.com/sites/default/files/mp3/bbriefing_2013..., which is an interview (starting at 1:36) with one of the public defenders involved.