Dudeney numbers

A Dudeney number is a positive integer that is a perfect cube such that the sum of its decimal digits is equal to the cube root of the number. There are exactly six such integers, i.e. $1,\, 512,\, 4913,\, 5832,\, 17576,$ and $19683:$

$\left\{ \begin{array}{llll} 1 = {\color{#3D99F6}1}^3 && {\color{#3D99F6}1} = 1\\ 512 = {\color{#3D99F6}8}^3 && {\color{#3D99F6}8} = 5 + 1 + 2\\ 4913 = {\color{#3D99F6}17}^3 && {\color{#3D99F6}17} = 4 + 9 + 1 + 3\\ 5832 = {\color{#3D99F6}18}^3 && {\color{#3D99F6}18} = 5 + 8 + 3 + 2\\ 17576 = {\color{#3D99F6}26}^3 && {\color{#3D99F6}26} = 1 + 7 + 5 + 7 + 6\\ 19683 = {\color{#3D99F6}27}^3 && {\color{#3D99F6}27} = 1 + 9 + 6 + 8 + 3. \end{array} \right.$

We can extend the definition to include not only perfect cubes but any perfect power. A $k$ -Dudeney number will therefore be any positive integer such that the sum of its decimal digits is equal to the $k^\text{th}$ root of the number.

Find $\displaystyle \sum_{k=2}^{19} \sqrt[k]{D_k},$ where $D_k$ is the biggest $k$ -Dudeney number.