Denesting of Ramanujan's radicals (2)

Since $9(2^{\frac13} - 1)(2^{\frac13}+1)^3 \; = \; 9(2^{\frac13}-1)(2 + 3\cdot2^{\frac23} + 3\cdot2^{\frac13} + 1) = 27(2^{\frac13}-1)(1+2^{\frac13}+2^{\frac23}) \; = \; 27(2-1) = 27 = (2+1)^3$ we have $9(2^{\frac13}-1) \; = \; \left(\frac{2+1}{2^{\frac13} + 1}\right)^3 \; = \; \big(1 - 2^{\frac13} + 2^{\frac23}\big)^3$ and hence $\sqrt[3]{\sqrt[3]{2}-1} \; =\; \sqrt[3]{\tfrac{1}{9}} - \sqrt[3]{\tfrac{2}{9}} + \sqrt[3]{\tfrac{4}{9}}$ and hence $k = \boxed{2}$ .