Simpler than it looks

$\begin{aligned} \log_{17} \left( \log_{11} \left( \sqrt{x+11} + \sqrt{x} \right) \right) & = 0 \\ \log_{11} \left( \sqrt{x+11} + \sqrt{x} \right) & = 1 \\ \sqrt{x+11} + \sqrt{x} & = 11 \\ x + 11 + 2 \sqrt{x^2+11x} + x & = 121 \\ 2 \sqrt{x^2+11x} & = 110 - 2x \\ \sqrt{x^2+11x} & = 55 - x \\ x^2+11x & = 55^2-110x + x^2 \\ 121x & = 55^2 \\ \implies x & = \boxed{25} \end{aligned}$

Since there is only one value of $x$ satisfying the equation, the sum of solutions is $\boxed{25}$ .

$log_{17}log_{11}(\sqrt(x+11) + \sqrt{x} ) = 0$

$\longrightarrow \large rise \ everything \ to \ the \ power \ of \ 17$

$log_{11}(\sqrt(x+11) + \sqrt{x} ) = 1$

$\longrightarrow rise \ everything \ to \ the \ power \ of \ 11$

$(\sqrt(x+11) + \sqrt{x} ) = 11$ --------- (i)

$\sqrt { x\quad +\quad 11 } =\quad 11\quad -\quad \sqrt { x } \quad$

${ (\sqrt { x\quad +\quad 11 } ) }^{ 2 }\quad =\quad (\quad 11\quad -\quad \sqrt { x } )^{ 2 }$

$\underline { x } +\quad 11\quad =\quad 121\quad +\quad \underline { x } \quad -\quad 22\quad \sqrt { x }$

${ (\frac { 110 }{ 22 } ) }^{ 2 }=\quad (\sqrt { x } )^{ 2 }\quad \Rrightarrow \quad x\quad =\quad 25\quad$

$Substituting \ x = 25$ in equation (i)

$\longrightarrow \ we \ will \ get \ 11 = 11$

$L.H.S = R.H.S$