Contrary to your claim, DNA strands do have sufficient periodicity to produce an x-ray diffraction pattern: in essence, when seen from the side, the helix collapses to two sets of diagonal lines, each of which acts as a diffraction grating, producing as an end result a cross formed from two diagonal sets of dots.
Moreover, this simple diffraction pattern can be readily produced at home using a laser pointer and a lightbulb filament, as seen in e.g. this question, the YouTube video it links to, and several related videos on YouTube, particularly DNA diffraction with a LASER! || MinuteLaboratory #14.
To scale up from a single molecule to an observable diffraction pattern of the double spiral, the clearest route is to crystallize the DNA, forming a periodic arrangement of DNA strands with the same orientation, so that the diffraction patterns from their helical structures adds up to an observable pattern. If you then make sure that the crystal structure is of identical DNA strands (with the same nucleotide sequences) and that these are synced up, then you can peer into the internal structure of the nucleic bases, but that's not necessary if, like Rosalind Franklin, all you're after is the diffraction pattern from the double helix.