TOC alignment in memoir

Question

I'm trying to get chapters and sections to line up in a memoir ToC as:

Chapter one        Chapter title
                   Section title
                   Another section
                   A third section

Chapter two        Second chapter title 
                   Another etc.

I really can't fathom the pseudo code on p.158 of the manual but I've tried setting widths for \cftpartnumwidth and \cftchapternumwidth but I can't get these boxes to work. My code:

% part
\cftpagenumbersoff{part}
\setlength{\cftpartnumwidth}{4.0cm}

% chapter
\setlength{\cftchapternumwidth}{4.0cm}
\renewcommand*{\cftchaptername}{Chapter\space}
\renewcommand*{\cftchapteraftersnum}{\hfill}
\renewcommand*{\chapternumberline}[1]{\cftchaptername \numtoname #1  \cftchapteraftersnum}

% section
\setlength{\cftsectionnumwidth}{4.0cm}

I can control the section title placement, but something is breaking the chapter title placement (I presume the renewal definition of \cftchapteraftersnum.

Can anyone give me the spell? thanks

============================== UPDATE:

I modified the code with Gonzalo's suggestion to:

\newlength\mylen
\settowidth\mylen{\bfseries\chaptername\hspace{3em}}

% part
\cftpagenumbersoff{part}
\addtolength\cftpartnumwidth{\mylen}

% chapter
\addtolength\cftchapternumwidth{\mylen}
\renewcommand\cftchapterpresnum{\chaptername~} 

% section
\renewcommand\cftsectionindent{\dimexpr\mylen+1.5em\relax}

But it gives this:

The "Sections" are not lining up with the "Chapters"...

Answer 1

The following code does what you want (change the value of the length \chapind according to your needs):

\documentclass{memoir}

\newlength\chapind
\setlength\chapind{2.8cm}% change according to your needs

\cftpagenumbersoff{part}
\setlength{\cftpartnumwidth}{\chapind}

\renewcommand*{\chapternumberline}[1]{%
\parbox[t]{\chapind}{\chaptername~\numtoname#1\cftchapteraftersnum}%
}

\renewcommand\cftsectionindent{\dimexpr\chapind-2.1em\relax}
\makeatletter
\def\cftsectionpresnum #1\@cftasnum{}
\makeatother

\begin{document}

\tableofcontents*

\part{Test Part One}
\chapter{Test Chapter One}
\section{Section One One}
\section{Section One Two}
\section{Section One Three}

\part{Test Part Two}
\chapter{Test Chapter Two}
\section{Section Two One}
\section{Section Two Two}
\section{Section Two Three}

\end{document}