How is it that earth's inner core is still solid? Earth's inner core is made up of an alloy of Iron and Nickel.

The important note here is that it is solid.

The temperature at the inner core is about $5,763K = 5,490\text{ degrees C}$.

Now the melting point of both solid iron and nickel is about $1,500\text{ degrees C}$, and even though they are combined, I still think that $5,490$ is much greater then the melting point of the alloy.

Regardless, how is it that the inner core still stays solid? I would expect that at such higher temperatures, a phase change to liquid, or maybe even gas would occur.

Why does it not?
The melting point of the vast majority of materials increases as more pressure is applied. At https://www.researchgate.net/figure/312054885_fig1_Figure-231-Experimental-pressure-temperature-phase-diagram-of-pure-iron-a is a phase diagram showing how this variation occurs for pure iron:


From https://physics.stackexchange.com/questions/184032/what-is-the-pressure-at-the-center-of-the-earth-or-a-neutron-star the pressure at the centre of the Earth is 365GPa. At this pressure we can see from the graph that the melting point of pure iron is well over 5000k. Thus, given the experimental uncertainties and the that the core is not pure iron, it is easily conceivable that the core of the Earth is solid, or at least a large proportion of it is.