Bass, your turn next on this one.

Part of the answer you are after is found in the way earthquake waves pass through the earth. The measurement of these sound waves is called refractive seismology. The theory that the center of the earth was mostly iron and nickle was able to be confirmed by finding the correct mathematical model of how the sounds traveled through the different layers of the earth and what they had to be made of to get the results observed. The theory was only provable when two technological developments came together. These were the development of very accurate clocks and the development of computers to both record the data and process the mathematical models.

The theory first came about due to the massive amount of nickle-iron meteorites compared to stony meteorites.

The rest of the answer probably is contained in the way molten earth materials differentiate themselves from one another. That is Bass's area of expertise so I will pass this question off to him. (Or that uppity student who is now taking earth materials - If they recently covered it he will be feeling cocky - and the answer will probably be more accurate.)