Aligning cases conditions from multiple nesting levels

I think I would do it in the way that egreg hinted at, and just use one dcases envrionment

Note that I have replaced your split environment with an array and used @{}c@{} to remove column space (check your original .log file- you should have got a warning that you weren't allowed to use split in that context).

MWE

\documentclass{article}
\usepackage{mathtools}

\begin{document}
\[
y_{ic}[n] = 
    \begin{dcases}
          y_0 + x_0 - x_0 (1 - k_2)^n & k_1 = 0 \\
          k_5 + \left( y_0 - k_5 \right) (1 - k_1)^n + x_0 k_1 n (1 - k_1)^{n-1} & k_1 \neq 0, k_1 = k_2 \\
          k_5 + \left( y_0 - k_5 \right) (1 - k_1)^n+ x_0 \frac{k_2}{k_1 - k_2} \left[ \begin{array}{@{}c@{}}(1 - k_2)^n \\ - (1 - k_1)^n \end{array} \right] & k_1 \neq 0, k_1 \neq k_2
    \end{dcases}
\]
\end{document}

If I were to do this, I would have listened to egreg's comments above. But the following works though coding may be ugly:

\documentclass{article}
\usepackage{mathtools,breqn}
\begin{document}
  \begin{dmath*}
y_{ic}[n] = \begin{dcases}
                          y_0 + x_0 - x_0 (1 - k_2)^n & k_1 = 0 \\
                          k_5 + \left( y_0 - k_5 \right) (1 - k_1)^n + \begin{dcases}
                                             x_0 k_1 n (1 - k_1)^{n-1}  &  \makebox[-11pt][l]{$k_1 \neq 0, k_1 = k_2$}\\
                                             x_0 \frac{k_2}{k_1 - k_2} \left[ \begin{aligned}(1 - k_2)^n \\  - (1 - k_1)^n \end{aligned} \right] & \makebox[-11pt][l]{$k_1 \neq 0, k_1 \neq k_2$}   
                                             \end{dcases}                                         
                  \end{dcases}
\end{dmath*}
\end{document}

Here are some more options that you could explore. Note that cases and friends can be easily emulated using an ordinary array:

\documentclass{article}
\begin{document}
\[
  \setbox1=\hbox{$k_1 \neq 0, k_1 = k_2$}
  y_{ic}[n] = \left\{\begin{array}{@{}l@{\quad}l}
    y_0 + x_0 - x_0 (1 - k_2)^n, & \hspace*{\dimexpr-1em-\wd1-\nulldelimiterspace}k_1 = 0 \\[2\jot]
    k_5 + (y_0 - k_5) (1 - k_1)^n + \left\{\begin{array}{@{}l@{\quad}l@{}}
      x_0 k_1 n (1 - k_1)^{n-1}, & k_1 \neq 0, k_1 = k_2 \\[\jot]
      \frac{x_0 k_2}{k_1 - k_2} [(1 - k_2)^n - (1 - k_1)^n], & k_1 \neq 0, k_1 \neq k_2
    \end{array}\right.
  \end{array}\right.
\]

\bigskip

\[
  \setbox1=\hbox{$k_1 \neq 0, k_1 = k_2$}
  y_{ic}[n] = \left\{\begin{array}{@{}l@{\quad}l}
    y_0 + x_0 - x_0 (1 - k_2)^n, & \hspace*{\dimexpr-1em-\wd1-\nulldelimiterspace}k_1 = 0 \\[2\jot]
    k_5 + (y_0 - k_5) (1 - k_1)^n \\ \qquad {} + \left\{\begin{array}{@{}l@{\quad}l@{}}
      x_0 k_1 n (1 - k_1)^{n-1}, & k_1 \neq 0, k_1 = k_2 \\[\jot]
      \frac{x_0 k_2}{k_1 - k_2} [(1 - k_2)^n - (1 - k_1)^n ], & k_1 \neq 0, k_1 \neq k_2
    \end{array}\right.
  \end{array}\right.
\]
\end{document}

The space-correction in the first row consists of 3 parts:

1. An em skip, stemming from the forced @{\quad} in the separation between the two columns;
2. \wd1 (or the width of \box1) equal to the width of the condition(s) in row 2 and 3;
3. \nulldelimiterspace, since using \right. actually inserts a small gap.