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First, let $\pi$ be some K-cycle on $[n] = \lbrace 1, \ldots, n \rbrace$
\[
\pi = \begin{pmatrix}
      1 & 2 & \ldots & k & k+1 & \ldots & n \\
      2 & 3 & \ldots & 1 & k+1 & \ldots & n
      \end{pmatrix}
\]
Let $\langle a, b \rangle$ be the transposition that switches the contents of $a$ and $b$.
By hypothesis $\pi$ is generated by DISTINCT switches on $[n]$.
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Introduce two ``new bodies'' $\lbrace x, y \rbrace$ and write 
\[
\pi^\star = \begin{pmatrix}
      1 & 2 & \ldots & k & k+1 & \ldots & n & x & y \\
      2 & 3 & \ldots & 1 & k+1 & \ldots & n & x & y
      \end{pmatrix}
\]
For any $i = 1, \ldots , k$ let $\sigma$ be the (L-to-R) series of switches
\[
\sigma = \left(\langle x, 1 \rangle \langle x, 2 \rangle \ldots \langle x, i \rangle \right)
		 \left( \langle y, i+1 \rangle \langle y, i+2 \rangle \ldots \langle y, k \rangle \right)
		 \left( \langle x, i+1 \rangle \right)
		 \left( \langle y, i \rangle \right)
\]
Note each switch exchanges an element of $[n]$ with one of $\lbrace x, y \rbrace$ so they're all distinct
from the switches within $[n]$ that generated $\pi$, and also from $\langle x, y \rangle$. By routine
verification,
\[
\pi^\star \sigma  = \begin{pmatrix} 1 & 2 & \ldots & n & x & y \\
                                    1 & 2 & \ldots & n & y & x \end{pmatrix}
\]
i.e. $\sigma$ inverts the K-cycle and leaves $x$ and $y$ switches (without performing $\langle x, y \rangle$).
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Now let $\pi$ be an ARBITRARY permutation on $[n]$: it consists of disjoint (nontrivial) cycles, and each can be
inverted as above in sequence, after which $x$ and $y$ can be switched if necessary via $\lbrace x, y \rbrace$,
as was desired
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