Note this cone is like the one in problem 9.1.

The sphere is inscribed in the cone with radius $1$, and has radius $\dfrac{\sqrt{3}}{3}$, so you can figure out the height of the cone (Spolier alert: it is the same as the height of the cone of problem 9.1!).

The other sphere will be smaller. Here is a side view diagram of the two spheres in the cone: