First, I agree with BTD partially. When he notes that Senator Kennedy's bill doesn't hold constitutional water, he's right on target that way in which Kennedy is micromanaging executive decisions will never be tolerated by the courts (if it were to make it pass Congress and then pass the veto). It is this language specifically that gets Kennedy's bill in trouble:

[N]o Federal funds may be obligated or expended by the United States Government to increase the number of United States forces in Iraq above the level for such forces which existed as of January 1, 2007, without a specific authorization of Congress by law for such an increase.

I think the problem has two parts. (1) This act impliedly contradicts the second AUMF which gave the executive just the kind of authority to adjust troop levels. Now, Congress certainly has the power to overturn its earlier acts, however, this bill doesn't seem to replace or modify the second AUMF. (I say "seem" because I'm having trouble finding the exact text of the proposal. I'm working from the Senator's website explanation of the legislation.)

What this means is that Congress, should the bill be passed and signed, would be giving the executive mixed signals. As a practical matter, that's not a good way to fight a war. As a legal matter, the courts will be asked to resolve the contradiction.

(2) That brings me to the second part. It's unconsitutional for the Congress to directly manage war because it's an usurpation of the executive's Article II power. Congress gets to appropriate money--that is devote it to a specific purpose. It can put whatever conditions and requirements it wants (mostly) before granting the appropriation, however, once appropriated, Congress' role becomes more limited, especially when it comes up against the constitutional powers of the executive.

Second, I disagree with BTD that Congress' only remedy to the Iraq war is to end it. I think it entirely consistent with Congress' constitutional powers to refuse to authorize all the funds that are requested by the executive for war. Congress can refuse to fund the executive. That's part of its enumerated powers under Article I sec. 8.

So, Congress can lower the levels of funds it gives the the executive to wage war. What it cannot do is tell the executive how to use the funds it authorizes for war. That's where it runs afoul of Article II. So, legislation giving the Administration only $40B rather than $80B in appropriations would be constitutional. Saying that the executive can spend no money on moving troops around, however would not.

In that way, Congress can impact the ability of the executive to wage war short of
"undeclaring" it.

Finally, in response to Andreas, there are no legally binding "Nuremberg principles." Nuremberg is fantastic precedent for International Military Tribunals. But it set no law on any of the countries involved.

For whatever reason, the myth of "Nuremberg principles" has been bandied around quite a bit the last few days. I saw it first in the reporting on the fellow who is currently facing a court-martial for refusing to deploy. Sadly, the sloppy reporting on that has perpetuated the myth of "Nuremberg principles."

The U.S. prosecutor at Nuremberg, Robert Jackson, emphasized that war crimes are war crimes no matter what country commits them. The United States more than any other sovereign state took the lead in the movement to generalize the principles underlying the Nuremberg judgment. At its initiative, the General Assembly of the United Nations, in 1945 unanimously affirmed "the principles of international law recognized by the Charter of the Nuremberg Tribunal" in Resolution 95(I). This resolution was an official action of governments. At the direction of the UN membership, the International Law Commission, a body of international law experts from all the principal legal systems in the world, formulated the Principles of Nuremberg in 1950. These seven Principles of International Law are printed in full in the adjoining box, to indicate the basic standards of international responsibility governing the commission of war crimes.
...
The substantive content of international law is contained in a series of international treaties that have been ratified by the United States, including principally the five Hague Conventions of 1907 and the four Geneva Conventions of 1949.

These international treaties are part of "the supreme law of the land" by virtue of Article VI of the U.S. Constitution. Customary rules of international law governing warfare are also applicable to the obligations of American citizens.