How to solve the Iraqi mess? Get others involved. Why would anyone else want to get involved you say? Here's the sides of this conflict:
1-Kurds vs Turks
2-Sunni vs Shia
3-Persian vs Arab
4-Fundamentalists vs secularists
I'll start with #1. The Kurds have had a defacto autonomy since the no-fly zones were enforced back in Clinton's presidency. The Kurdish people have a presence in Syria, Iran and Turkey in addition to their main area of northern Iraq. The place where the Kurds have the most conflict is with the Turks. This is an ancient conflict. On the Kurds' side there are two movements, one that is amenable to being somewhat peaceful with the Turks but they cannot be buddies with the Turks, and the other movement is dedicated toward conflict with the Turks. Because of this conflict, we can drag the Turks into helping sort out the northern third of Irag.
2-Sunnis and Shias don't like each other. That's an understatement...LOL! Think Protestant vs Catholic times ten. As much intolerance as these monotheistic religions have unleashed upon the rest of the world, there is a special place in their dark hearts for those they consider heretics. Iraq has a majority of Shias. The Sunnis ran the place for centuries. Other than Iran (formerly Persia), the nations bordering Iraq are Sunni majority nations. They have a vested interest in not seeing a Shia radical regime arise. Let that be a motive for them to contribute troops, money and resources toward pacifying Iraq.
3-Persians vs Arabs predates Muhammed, Jesus Christ and Abraham. Heck, Persian vs everyone else in the world was part and parcel of the most ancient times! Not all Shia Arabs are particularly well-disposed toward Iran. The Persian enemy is the OLDEST enemy of all. Let that be the wedge that keeps all of the majority Shia from unifying against the rest of everyone in this furball. Arab nations reinforcing pro-Arab Shia is doable if it's seen as trumping the religious conflict of Sunni vs Shia. The Saudis are not the only Arabs with pieces on the board, so see about dragging in the Gulf states to help the cause.
4-Fundamentalists vs secularists is another conflict. The secular regimes and royal houses don't want to see fundamentalism rage out of control as their grip on the lever of power would be broken. Whether the fundamentalism is Sunni-oriented as in the case of the Taliban and Al Queda or Shia-oriented as in the case of the Iraqi Shia who would wish to see a theocracy setting up a Caliphate in Baghdad, it's ALL BAD for all secular regimes involved including ones who do not border Iraq but who are vulnerable to the forces of fundamentalism, especially Egypt. Get the Egyptians involved!
It will be cheaper to pay the denizens of the Middle East to do the dirty work of stabilizing Iraq than to use our own troops and equipment. We can pay a smaller bill and get the rich oil producing nations to pony up since they are flush with cash by using adroit diplomacy. It is in these other nations' interest to get involved but we have to show them what to do, how to do it and facilitate the process.
I could say a LOT more but I'll keep it down to this so I don't bore the hell out of you guys...LOL!
Rick