This task is a paradox. Fortunately, I like paradoxes and hence I will attempt to logically solve that which is entirely non-logical in hopes that I will reach a conclusion.

Step 1: Breaking down the paradox

"Find": to locate either temporally or spatially, or both.
"the exact middle": a temporal or spatial location equidistant from both ends of the segment, assuming there is one.
"of nowhere": a spatial location being non-existent and referring to an absence of specified space.

Step 2: Understanding the newly broken down nonsense

We can now understand that "nowhere" is qualified solely by its lack of 'whereness.' Hence its defined boundaries are that of not having a location, or a beginning and end. Hence it is an abstract term which refers to nothing finite, rather, is based solely upon the lack-there-of.
With this as our starting point, we now attempt to find an end point. This end would be the conclusion of the task, namely, finding the exact middle of nowhere. We now have a beginning and end - two segments which we shall refer to in our attempt to locate the middle.
Since we are looking for the exact middle we must establish some sort of measurement for our task. As it took roughly 10 seconds to read and understand the task, then another 30 to formulate a logical approach to the task, and another minute or so to type the entire thing out, we have approximately 2 minutes of elapsed time. This will be our scale.

A little diagram:
Beginning ---------------> Exact Middle ----------------> End
Breaking down ---------> Making sense ---------------> Conclusion
0 seconds ---------------> 1 minute --------------------> 2 minutes

Yet upon looking at our diagram it becomes clear that we were making sense of the paradox from the beginning, and hence the exact middle will be some point within the act of making sense, rather than the collective mental task.
This poses a new problem and one which cannot be resolved at the moment. Therefore we will return to making sense of nothing in order to procrastinate this fundamental dilemma.

Given that we have already established the qualities of "nowhere" and the meaning of "the exact middle," this leaves us with making sense of the verb "find." We assume that we are instructed to locate something, but given that we have the certainty of locating nothing, we must look elsewhere. Since we fundamentally cannot "find the exact middle of nowhere," we assume that the question is bound by laws which we must violate in order to succeed. Since "nowhere" is defined by an absence of 'whereness,' and this 'whereness' is the quality necessary to establish a middle, we conclude that the answer to the question is elsewhere.

The exact middle of nowhere is elsewhere.

Next question.