Here's the easy way: knowing the length, l, of string used, calculate its volume as [itex]\pi s^2l[/itex]. The volume of the ball is [itex](4/3)\pi B^3[/itex]. The "amount" of string is [itex]\pi s^2l[/itex] and the "amount" of air is [itex](4/3)\pi B^3- \pi s^2l[/itex].
I was looking for an (approximate) relation between B, s and l so that only the variables B and s would be needed to solve the problem. The string would be wound so as to minimize the space within.
I would build the sphere layer by layer, in each layer the string spiraling out to form a circle with small thickness. Every layer follow the "grooves" of the spiral in the lower one.