What impact has the Muslim resurgence had on world history? Why has the West been unable to cope with this development in a realistic manner?
It's important to remember that Islam has been a major world religion for quite some time, so the term resurgence does not really refer to membership, or even the mainstream beliefs of the vast majority of Muslims. The term generally refers to the resurgence of radical Islam, especially along Wahhabi lines such as al-Qaeda.
The impact on world history is most obvious in the military conflicts in the Near East, most notably in Afghanistan, and to a lesser degree in Iraq. Radical Islam provoked the invasion of Afghanistan after the 9/11 attacks, the pursuit of bin Laden and al-Qaeda around the globe, and that conflict widened to include attacks on US forces in Iraq as well. The impact then, in these cases, was to force or provoke Western society into investing massive military and financial resources into responding directly to some elements of the Muslim Resurgence.
The West has been largely unable to cope with this radicalization because, and this is a matter of opinion, it has fundamentally misunderstood or even ignored the root causes of the Resurgence. It has essentially declared war on a set of ideas. Terrorism has no military solution, and probably no diplomatic one. The roots and recruiting grounds of radical Islam are economic, and depend on crushing poverty to be successful. This is both inconvenient and complicated, with no short term solution, something the West, and the US in particular, does not typically handle well in recent years.
There are some issues within the question that might require some level of clarification. The notion of "resurgence" needs a bit more detail. Exactly how has the religion experienced a "resurgence?" It is a landmark religion of the West and along with Judaism and Christianity helps to establish "The Big Three" of Western Religion. I guess I need a bit more detail on this resurgent element. At what point was the religion dormant? The second issue brought out is the notion that the West has been unable to cope with this development. The religion of Islam has been around for such a long time that it seems that the religion is not subjected to experiences of it being stamped out or outward religious quests to eradicate it. I concede that there has been a great deal of challenge in trying to understand Islam in the modern sense with the rise of a radicalist presentation of it, but I don't think that this constitutes an "inability" to cope with it. If the question here is geared towards the radicalist interpretation, the argument can be made that radicalist Christianity or Judaism has been understood in the same way, as few can discern and "cope" with the rise of these factions, as well.
This is really a very loaded question and the answer to it really depends on your political views.
First of all, you could say that the West has been unable to cope with this development because the West is used to think of Muslims from a colonialist point of view. They are supposed to do what we tell them because they live in countries that are less developed and which we in the West have colonized. Because of this kind of attitude, the West does not deal with Muslims and Muslim countries as equals.
However, this is an anti-Western point of view and you can just as easily argue that the West has not been able to cope because the Muslim world is operating with a totally skewed view of the West. The Muslim world, you can argue, looks at actions that the West takes and invariably denounces them as anti-Muslim. They fail to understand the reasons and motives behind Western behavior. Because of this, they do not deal with the West in any reasonable way and that prevents the West from coping with them.