how to compute a discrete fourier transform on fragmented data
Basically, it is impossible to calculate the full DFT of your data. Take a look at the DFT matrix. 



All of its elements are non-zeroes. Thus, each and every signal element is important for any frequency. 

Take a look at this another way, had it been possible to do what you want, you could find the DFT of the signal, do an inverse DFT transform, and get the full signal back! Obviously, this is wrong. You don't have this information.

Nevertheless, it is possible to compute the frequencies of your data approximately. Recall that one of the possible explanations of DFT transform is that each of its coefficients is the best fit to a sum of shifted sine waves, $A* sin(w (x-x_0))$, or $a sin(wx)+b cos(wx)$, where $w$ is dependent on the coefficient index. Thus, you should do a least-squares fit to $a_i,b_i$, which are the coefficients of $a+bi$ in DFT.