I don't know minitoc well, but here's a way to obtain the desired formatting for the partial ToCs using the titletoc package; once all the settings have been defined in the preamble, all the job is done using a single \PartialToC command:
\documentclass{book}
\usepackage{xcolor}
\usepackage{mdframed}
\usepackage{titletoc}
\definecolor{secnum}{RGB}{13,151,225}
\definecolor{ptcbackground}{RGB}{212,237,252}
\definecolor{ptctitle}{RGB}{0,177,235}
\titlecontents{lsection}
[5.8em]{\sffamily}
{\color{secnum}\contentslabel{2.3em}\normalcolor}{}
{\titlerule*[1000pc]{.}\contentspage\\\hspace*{-5.8em}\vspace*{5pt}%
\color{white}\rule{\dimexpr\textwidth-15.5pt\relax}{1pt}}
\newcommand\PartialToC{%
\startcontents[chapters]%
\begin{mdframed}[backgroundcolor=ptcbackground,hidealllines=true]
\printcontents[chapters]{l}{1}{\colorbox{ptctitle}{%
\parbox[t]{\dimexpr\textwidth-2\fboxsep\relax}{%
\strut\color{white}\bfseries\sffamily\makebox[5em]{%
Chapter~\thechapter\hfill}Contents}}\vskip5pt}
\end{mdframed}%
}
\begin{document}
\tableofcontents
\chapter{Chapter One}
\PartialToC
\section{Section One One}
\section{Section One Two}
\section{Section One Three}
\section{Section One Four}
\chapter{Chapter Two}
\PartialToC
\section{Section Two One}
\section{Section Two Two}
\section{Section Three Three}
\end{document}
Some images of the partial ToCs:

and

In a comment to the answer has been requested to produce a similar formatting for the general table of contents; in this case, some additional works has to be done:
\documentclass{book}
\usepackage{xcolor}
\usepackage{mdframed}
\usepackage{titletoc}
\usepackage{etoolbox}
\definecolor{secnum}{RGB}{13,151,225}
\definecolor{ptcbackground}{RGB}{212,237,252}
\definecolor{ptctitle}{RGB}{0,177,235}
\pretocmd{\tableofcontents}{\begin{mdframed}[backgroundcolor=ptcbackground,hidealllines=true]}{}{}
\apptocmd{\tableofcontents}{\end{mdframed}}{}{}
\patchcmd{\tableofcontents}{\contentsname}{\color{ptctitle}\contentsname}{}{}
\titlecontents{section}
[4em]{\sffamily}
{\color{secnum}\contentslabel{2.3em}\normalcolor}{}
{\titlerule*[1000pc]{.}\contentspage\\\hspace*{-3em}\vspace*{2pt}%
\color{white}\rule{\dimexpr\textwidth-20pt\relax}{1pt}}
\titlecontents{lsection}
[5.8em]{\sffamily}
{\color{secnum}\contentslabel{2.3em}\normalcolor}{}
{\titlerule*[1000pc]{.}\contentspage\\\hspace*{-5.8em}\vspace*{2pt}%
\color{white}\rule{\dimexpr\textwidth-15.5pt\relax}{1pt}}
\makeatletter
\renewcommand*\l@chapter[2]{%
\ifnum \c@tocdepth >\m@ne
\addpenalty{-\@highpenalty}%
\vskip 1.0em \@plus\p@
\setlength\@tempdima{1.5em}%
\begingroup
\parindent \z@ \rightskip \@pnumwidth
\parfillskip -\@pnumwidth
\leavevmode
\advance\leftskip\@tempdima
\hskip -\leftskip
\colorbox{ptctitle}{\strut%
\makebox[\dimexpr\textwidth-2\fboxsep-7pt\relax][l]{%
\color{white}\bfseries\sffamily#1%
\nobreak\hfill\nobreak\hb@xt@\@pnumwidth{\hss #2}}}\par\smallskip
\penalty\@highpenalty
\endgroup
\fi}
\makeatother
\newcommand\PartialToC{%
\startcontents[chapters]%
\begin{mdframed}[backgroundcolor=ptcbackground,hidealllines=true]
\printcontents[chapters]{l}{1}{\colorbox{ptctitle}{%
\parbox[t]{\dimexpr\textwidth-2\fboxsep\relax}{%
\strut\color{white}\bfseries\sffamily\makebox[5em]{%
Chapter~\thechapter\hfill}Contents}}\vskip5pt}
\end{mdframed}%
}
\begin{document}
\tableofcontents
\chapter{Chapter One}
\PartialToC
\section{Section One One}
\section{Section One Two}
\section{Section One Three}
\section{Section One Four}
\stopcontents[chapters]
\chapter{Chapter Two}
\PartialToC
\section{Section Two One}
\section{Section Two Two}
\section{Section Two Three}
\stopcontents[chapters]
\chapter{Chapter Three}
\PartialToC
\section{Section Three One}
\section{Section Three Two}
\section{Section Three Three}
\section{Section Three Four}
\section{Section Three Five}
\end{document}
An image of the general ToC:

An image of one of the partial ToCs:

\documentclassand the appropriate packages that sets up the problem. While solving problems is fun, setting them up is not. Then those trying to help can simply cut and paste your MWE and get started on solving problem. – Peter Grill Nov 18 '12 at 7:04