You do no say what information you know and do not know. For example if the cube sinks and $h_1$ is big enough, it is possible that $s_2=0$. But if you know $s_1$ and $s_2$ then it is easy.

The volume of liquid displaced is $(s_1+s_2)n^2$ so the extra height (ignoring overflows) is $\dfrac{(s_1+s_2)n^2}{\pi R^2 - n^2}.$ So the final overall height of the liquid is $$\min\left(H,h_1+h_2+\dfrac{(s_1+s_2)n^2}{\pi R^2 - n^2}\right)$$ and if you know $s_1$ and $s_2$ then you do not need to use the densities.

It is possible to calculate $s_1$ and $s_2$ using the information available including the densities, and that is where there are several cases to consider.

You do no say what information you know and do not know. For example if the cube sinks and $h_1$ is big enough, it is possible that $s_2=0$. But if you know $s_1$ and $s_2$ then it is easy.

The volume of liquid displaced is $(s_1+s_2)n^2$ so the extra height (ignoring overflows) is $\dfrac{(s_1+s_2)n^2}{\pi R^2 }.$ So the final overall height of the liquid is $$\min\left(H,h_1+h_2+\dfrac{(s_1+s_2)n^2}{\pi R^2 }\right)$$ and if you know $s_1$ and $s_2$ then you do not need to use the densities.

It is possible to calculate $s_1$ and $s_2$ using the information available including the densities, and that is where there are several cases to consider.