How much silicon is in the Earth's core, and how did it get there?
The only way presently is to simulate the conditions in the core, as has been performed in the article Oxygen and silicon contents of Earth's core from high pressure metal–silicate
partitioning experiments (Ricolleau et al. 2011), whose results indicate that

that the oxygen metal–silicate partition coefficient increases
with increasing oxygen fugacity, temperature and pressure, whereas the silicon metal-silicate partition coefficient increases with decreasing oxygen fugacity, increasing temperature and pressure.

The model also revealing that the approximate concentration of silicon in the core (outer core) is up to 11 wt%, depending on what model of core formation was used in the simulation.  These results

in the core could satisfies the seismologically observed range of outer core
density deficits.

In terms of how long the silicon is believed to have been in the core, in the article Silicon in the Earth’s core (Georg et al. 2007), similarities with the silicon content with the Moon led to their conclusion that

$Si$ was already incorporated as a light element in the Earth’s core before the Moon formed.