The DoJ laid its arguments out pretty well in Assistant Attorney General's Letters to Senate Intelligence Committee, and the AUMF is an integral part of some justifications asserted therein.
I certainly prefer the more precise descriptions of "within the inherent power of the executive as augmented granted by the AUMF" to the less precise labels of "legal" and "illegal." And in the broad scheme of things, having some give and take, some tension between the branches of government, is something that I think is good for the Republic.
There is an opposite point of view, that relatively secret (and therefore not entirely, but relatively without oversight) executive actions are entirely appropriate; and the existence of actions undertaken thereby should be forever hidden from public scrutiny. No trials, no hearings, just "secret" action.
Is this not, however, the de facto current situation, that no persons who know what the President has asked for by way of surveillance have power to legally bring scrutiny upon these actions? What are the possibilities -- impeachment or judicial decree? Spilling the beans to Congress as a whole, or filing an objection with the US Supreme Court, is only going to get you slammed in jail.