Voice split across staves?

[Joel C. Salomon]

I’m trying to re-create John Crook’s original score to Peter Pan (yes,
it’s in the public domain) for the Mutopia project, and I’m finding that
my lack of musical education is getting in the way.

I’m trying to set “Pipe with the Ostrich”, as visible at
<https://books.google.com/books?id=J-wQAAAAYAAJ&pg=PT43>; my current
effort is at
<https://github.com/jcsalomon/MutopiaProject/blob/peterpan/ftp/CrookJ/PeterPan/PP06_Pipe_with_the_Ostrich/PP06_Pipe_with_the_Ostrich.ly>.

While critiques of my Lilypond style are also welcome, my specific
question is how best to show what seems to be a voice split across
staves.  The following example should show what I mean: I’ve set the
piece’s first bar twice, the first time as seems logical to me, the
second time as it appears in the printed score:

    \version "2.18.2"
    \language "english"
    global = {
      \key d \major
      \time 2/4
    }

    upperStaff = \relative c'' {
      d8.         e16 d8      b8      |
    << { \voiceOne
      d8.         e16 d8      b8      |
    } \new Voice { \voiceTwo
      fs4             fs8     fs      |
    } >>
    }

    lowerStaff = \relative c {
      <d a' fs'>4     q8      q       |
      <d a'>4         q8      q       |
    }

    \score {
      \context PianoStaff <<
        \new Staff = "upper" { \clef treble \global \upperStaff }
        \new Staff = "lower" { \clef bass \global \lowerStaff }
      >>
      \layout{ }
      \midi { }
    }

The “Cross-staff stems” section in the documentation shows a way to draw
the chords so they cross the gap, but remain connected. One one hand,
this is different from the printed source; on the other hand, this might
be better style. On the gripping hand, I can’t see how to apply the
sample code to music organized as I’ve been doing it.

Could somebody please point me in the right direction?

—Joel C. Salomon