Stretch a Tikz tree

You can selectively increase the level distance by saying e.g.

where level=2{l+=0.5cm,fork sep=8mm}{}, 

where I also adjusted the fork sep for the optics. MWE:

\documentclass[border=10pt]{standalone}
 \usepackage[edges]{forest}

\begin{document}


\begin{forest}
        for tree={
            thick,
            l sep=0.55cm,
            where level=2{l+=0.5cm,fork sep=8mm}{},         
            s sep=0.55cm,
            child anchor=west,
            parent anchor=east,
            grow'=east,
            draw,
            anchor=west,
            node options={draw,font=\sffamily},
            edge={semithick},
            align=center,
            node options={draw,font=\sffamily},
            edge={semithick},
            forked edges,
            where n children={11}{rotate=90, anchor=center}{},
            where n children=2{rotate=90, anchor=center}{},
            where n children=6{rotate=90, anchor=center}{},
            where n children=8{rotate=90, anchor=center}{},
        },
        [Mother branch, 
        [First Child
        [Grand child []]
        [Grand child    []]
        [Grand child []]
        [Grand child []]
        [Grand child []]
        [Grand child []]
        [Grand child []]
        [Grand child []]
        [Grand child []]
        [Grand child []]
        [Grand child []]
        ]
        [Second Child
        [Grand child [ ]]
        [Grand child [] ]
        [Grand child []]
        [Grand child []]
        [Grand child []]
        [Grand child []]
        [Grand child []]
        [Grand child []]
        ]
        ]
    \end{forest}
\end{document}

i'm guessing, that you looking for something like this:

which is obtained with the following mwe:

\documentclass[border=10pt]{standalone}
\usepackage[edges]{forest}

\begin{document}
\begin{forest}
        for tree={
            grow'=east,
            anchor=west,
            node options={draw, thick, font=\sffamily},
            edge={semithick},
            forked edges,
            l sep=6mm,
            s sep=4mm,
            fork sep = 2.5mm,           % new, distance from parent to branching point
            where level=1{s sep=1mm}{}, % new
            where level=2{}{rotate=90, anchor=center} % new
        },
        [Mother branch,
        [First Child
        [Grand child []]
        [Grand child    []]
        [Grand child []]
        [Grand child []]
        [Grand child []]
        [Grand child []]
        [Grand child []]
        [Grand child []]
        [Grand child []]
        [Grand child []]
        [Grand child []]
        ]
        [Second Child
        [Grand child [ ]]
        [Grand child [] ]
        [Grand child []]
        [Grand child []]
        [Grand child []]
        [Grand child []]
        [Grand child []]
        [Grand child []]
        ]
        ]
    \end{forest}
\end{document}

changes in comparison to your mwe are indicated with % new. i also remove all doubled code lines in your mwe.