Below there's one possible solution; the code includes 1) A redefinition of \@chapter (as defined in report.cls) to include the word "Chapter" in the chapter entries of the ToC. 2) Two newly defined commands: \bappendix and \eappendix; the former starts a group, adds "Appendices" to the ToC and redefines \section (as defined in report.cls) to modify the section counter as requested; the latter simply ends the group:
\documentclass{report}
\newcounter{appendix}[chapter]
\makeatletter
\def\@chapter[#1]#2{\ifnum \c@secnumdepth >\m@ne
\refstepcounter{chapter}%
\typeout{\@chapapp\space\thechapter.}%
\addcontentsline{toc}{chapter}%
{\@chapapp~\thechapter: #1}%
\else
\addcontentsline{toc}{chapter}{#1}%
\fi
\chaptermark{#1}%
\addtocontents{lof}{\protect\addvspace{10\p@}}%
\addtocontents{lot}{\protect\addvspace{10\p@}}%
\if@twocolumn
\@topnewpage[\@makechapterhead{#2}]%
\else
\@makechapterhead{#2}%
\@afterheading
\fi}
\newcommand\bappendix{%
\addtocontents{toc}{\protect\addvspace{10pt}Appendices}
\begingroup
\renewcommand\section{\stepcounter{appendix}%
\renewcommand\thesection{\thechapter.\Alph{appendix}}
\@startsection {section}{1}{\z@}%
{-3.5ex \@plus -1ex \@minus -.2ex}%
{2.3ex \@plus.2ex}%
{\normalfont\Large\bfseries}}}
\makeatother
\newcommand\eappendix{\endgroup}
\begin{document}
\tableofcontents
\chapter{First Chapter}
\section{Section One One}
\section{Section One Two}
\bappendix
\section{Appendix One A}
\section{Appendix One B}
\eappendix
\chapter{Second Chapter}
\section{Section Two One}
\section{Section Two Two}
\bappendix
\section{Appendix Two A}
\section{Appendix Two B}
\eappendix
\end{document}

Here's the same approach, but using the etoolbox package to simplify the code:
\documentclass{report}
\usepackage{etoolbox}
\newcounter{appendix}[chapter]
\makeatletter
\patchcmd{\@chapter}{\protect\numberline{\thechapter}#1}
{\@chapapp~\thechapter: #1}{}{}
\newcommand\bappendix{%
\addtocontents{toc}{\protect\addvspace{10pt}Appendices}
\begingroup
\pretocmd{\section}{\stepcounter{appendix}%
\renewcommand\thesection{\thechapter.\Alph{appendix}}}{}{}}
\makeatother
\newcommand\eappendix{\endgroup}
\begin{document}
\tableofcontents
\chapter{First Chapter}
\section{Section One One}
\section{Section One Two}
\bappendix
\section{Appendix One A}
\section{Appendix One B}
\eappendix
\chapter{Second Chapter}
\section{Section Two One}
\section{Section Two Two}
\bappendix
\section{Appendix Two A}
\section{Appendix Two B}
\eappendix
\end{document}
Here's now a much more simple solution using the subappendices environment from the appendix package; the subappendices environment was slightly modified (to add some vertical space before "Appendices" in the ToC) using the etoolbox package; this package was also used to add "Chapter" before the chapter entries in the ToC:
\documentclass{report}
\usepackage{appendix}
\usepackage{etoolbox}
\AtBeginEnvironment{subappendices}{%
\addtocontents{toc}{\protect\addvspace{10pt}Appendices}
}
\makeatletter
\patchcmd{\@chapter}{\protect\numberline{\thechapter}#1}
{\@chapapp~\thechapter: #1}{}{}
\makeatother
\begin{document}
\tableofcontents
\chapter{First Chapter}
\section{Section One One}
\section{Section One Two}
\begin{subappendices}
\section{Appendix One A}
\section{Appendix One B}
\end{subappendices}
\chapter{Second Chapter}
\section{Section Two One}
\section{Section Two Two}
\begin{subappendices}
\section{Appendix Two A}
\section{Appendix Two B}
\end{subappendices}
\end{document}