Data for the Fresnel image analysis of Figure 5 of "Formations of Narrow Stripes and Vortex-Antivortex Pairs in a Quasi-Two-Dimensional Ferromagnet K2CuF4" by Yoshihiko Togawa, Tetsuya Akashi, Hiroto Kasai, Gary W. Paterson, Stephen McVitie, Yusuke Kousaka, Hiroyuki Shinada, Jun-ichiro Kishine, and Jun Akimitsu.