You're on the right track with seismology, but the reflection of certain seismic waves only prove evidence of the phase state (solid/liquid) of Earth's internal layers rather than what they are made of. The fact that shear waves do not propagate through the outer core show that it is molten. Reflected pressure waves from the outer core do so at a certain angle due to not just the compositional change from silicates to iron, but physical change from solid to fluid, the angle resulting in a shadow zone for where the reflected waves appear back at the Earth's surface. The core shadow-zone and its interpretation in terms of a fluid core were well established in 1936 when Inge Lehmann (a Danish seismologist) reported weak P-wave arrivals within the shadow zone. She interpreted these in terms of an inner core with higher seismic velocity. However, the existence of the inner core remained controversial for many years. Improved seismometer design, digital signal treatment and the setting up of seismic arrays have provided corroborating evidence. The existence of a solid inner core is also supported by analyses of the Earth’s natural vibrations.

Interestingly, ideas of the core being made of iron were put forth long before seismology got that far. It was known from the study of meteorites that some had a rock-like composition, while others were much denser, composed largely of iron. In 1897 E. Wiechert, who subsequently became a renowned German seismologist, suggested that the interior of the Earth might consist of a dense metallic core, cloaked in a rocky outer cover. He called this cloak the “Mantel,” which later became anglicized to mantle. When Wiechert made the iron-core hypothesis, it had already been known for 100 years that the Earth had a lot more mass inside than it did near the surface, so it wasn't completely out of the blue to equate meteorite iron with a planetary interior.