Császára et al. J. Chem. Phys. 122, 214305 (2005) has a nice table of determination of the bond angle of water per year (missing probably Linus Pauling first predicting 90° from approximations in quantum mechanics in 1931).
The earliest measurement cited is E. K. Plyler, Phys. Rev. 39, 77 (1932). In this paper, the angle is derived from the infrared absorption frequencies of water. The result was still close but not exact, providing an angle of 115°.
The next one had to wait to 1945, printed in the book G. Herzberg, Molecular Spectra and Molecular Structure Vol. II (1945) that was for decades the reference for the angle. The results come from the analysis of a series of experiments (mostly published in German) by R. Mecke and coworkers from 1933 to 1942. The angle cited is 104.45° (which coincides already with your image). Again from infrared spectroscopy.
Note that all values cited are a mixture of experiments and inferring from a preexisting theory as it is very hard to "take a direct picture" of a single molecule.