Forget the string and think of the lead and wood just below the surface. The total buoyant force on the lead and wood must be equal to the weight of the lead and wood. The buoyant force is the weight of the displaced water; the volume of the displaced water is the same as the volume of the lead...
To find k: You are given the total charge (Q) and the charge density. Integrate the charge density over the annulus to get an expression for the total charge. Equate this with Q and solve for k.