$\begin{array}{l}
\sqrt {6 + 2\sqrt {5 - \sqrt {13 + 4\sqrt 3 } } } \\
= \sqrt {6 + 2\sqrt {5 - \sqrt {12 + 1 + 2\sqrt {12} } } } \\
= \sqrt {6 + 2\sqrt {5 - \sqrt {{{\left( {\sqrt {12} + 1} \right)}^2}} } } \\
= \sqrt {6 + 2\sqrt {5 - \sqrt {12} - 1} } \\
= \sqrt {6 + 2\sqrt {4 - 2\sqrt 3 } } \\
= \sqrt {6 + 2\sqrt {3 + 1 - 2\sqrt 3 } } \\
= \sqrt {6 + 2\sqrt {{{\left( {\sqrt 3 - 1} \right)}^2}} } \\
= \sqrt {6 + 2\left( {\sqrt 3 - 1} \right)} \\
= \sqrt {6 + 2\sqrt 3 - 2} \\
= \sqrt {4 + 2\sqrt 3 } \\
= \sqrt {3 + 1 + 2\sqrt 3 } \\
= \sqrt {{{\left( {\sqrt 3 + 1} \right)}^2}} \\
= \sqrt 3 + 1
\end{array}$