Where does $\frac{N}{2}$ came from in approximating an N-point DFT?
The common formulation of the forward DFT preserves energy (Parseval's theorem).  This means that a longer constant magnitude sinewave input to a DFT, which has proportionally more energy, must be represented by a proportionally larger value in the DFT result, preserving that greater energy.  Thus a factor of N, which scales with the length of the input.

For strictly real input, this energy is divided evenly between the positive frequency bin (k) and the negative frequency bin (N-k) of a DFT result (for a DFT of length N). These two complex result bins are always complex conjugates of each other (thus equal in magnitude) for strictly real input to a DFT.  Thus another factor of 1/2 is included to do this even energy split.

N and 1/2 results in a factor of N/2 which has to appear somewhere in the result of an energy preserving formulation of a forward DFT.

Note that the 1/2 does not appear for bin 0 or for bin N/2 of even length DFTs, as these have no complex conjugate image (or rather they include both of their complex conjugates summed together into a single result bin).

Also note that this Sinc function result is an approximation.  The actual result is closer to the sum of two Dirichlet functions (also called periodic Sinc functions).  See here for more details.