Merely shutting down the murder factories will, in time, be enough.
As 9/11 or London (or Madrid, or Sharm el-Sheik for that matter) didn't originate from our late invasion of Iraq and the overthrow of Saddam, neither did it originate with the formation of the nation of Israel. The bloody origins go back a lot further than the Madrassahs phenomenon, of course. They go back to the Prophet Mohammad himself, his examples and teachings, and those adherents who immediately followed him which all set the bloody pattern for Islam. In other words, the cancer dates back over 1300 years and has never been successfully exorcised despite whatever has been tried against it.
So, the primary malignancy is Islam itself, which has proved itself quite capable of mountains of atrocities piled upon atrocities over the centuries - not dependent on any particular set of individual actors or political situations.
I doubt the long term situation will change by closing down the "murder factories" (as indeed they are, and their closing down will be a vast, if temporary, improvement). The lasting change can only come when these people have a complete change of heart, not a costume change of Islamic clothes for other Islamic garments.
Quite frankly, I think Christianity is the only power capable of doing that job effectively to the deep extent required. After all, one is not here addressing a mere nation state, emperor, race, ethnicity or even Axis of Evil. What is involved at bottom is a seamless and millenium-long world-view of immense culture and societal power which has permanently changed vast regions of the globe and completely erased previously thriving cultures.
To expand the theme further, this is a spiritual as well as a temporal war - perhaps primarily spiritual. Therefore, the cure must also be spiritual (to loosely quote Douglas MacArthur). The Japanese Empire, Nazi Germany and even the Soviet Union were regional, late-blooming pikers compared to Islam.
We can aid the necessary change by promoting personal, civil and religious freedoms within these miserable, bound countries - such as we are attempting to do in Iraq and Afghanistan. But we cannot force it to happen. It goes down deep to the very core cultural and theological beliefs of large societies and individuals who comprise in excess of a quarter of the world's population. This is not something simply grafted or veneered on top of a single society. But the cosmic change will not happen if Islam is left alone on it's own to fester along in it's own malignant soup of theology, precedent and Sharia Law - even without the "murder factories". As you indicate, it will require strong temporal support on our part and it is not a quick fix. It will not be done alone with armies.
In the end, IMHO, only the intercession of the Holy Spirit in uncounted legions of lives of current Muslims and their offspring will make the necessary change to make it work and deliver us all from the scourge of this horror. That is ultimately beyond our immediate control, but not beyond our hopes and prayers aided by our long-term intelligent temporal actions.
It will take a lot more years than you and I have, Congressman, and certainly more power and wisdom than can be packed into 750 words. ;>)
save for later, but...
Passive solutions will not work.