Fock operator invariance under unitary transformation

I know how to show that the Coulomb operator of the Fock operator is invariant under a unitary transformation of the orbitals, as on page 121 of Szabo and Ostlund, but the indices in my proof for the exchange operator are just not working. What might I be missing?
I have 


\begin{align}
\sum\_i\hat{K}\_i'(1) &=\sum\_i\int dr\_2\chi\_i'^\*(2)\chi\_j'(2)/r\_{12} \\
 &=\sum\_i\int dr\_2\sum\_kU\_{ki}^\*\chi\_k^\*(2)\sum\_lU\_{lj}\chi\_l(2)/r\_{12}\\
 &=\sum\_{ikl}\int dr\_2U\_{ki}^\*U\_{lj}\chi\_k^\*(2)\chi\_l(2)/r\_{12}\\
 &=\sum\_{i}\int dr\_2\chi\_i^\*(2)\chi\_i(2)/r\_{12}\\
 &\neq\sum\_{i}\int dr\_2\chi\_i^\*(2)\chi\_j(2)/r\_{12}.
\end{align}




The exchange operator is not a operator by itself, it is only defined with the orbital it is working on: 
$%
\newcommand{\ll}{\left\langle}\newcommand{\rr}{\right\rangle}
\newcommand{\lb}{\left|}\newcommand{\rb}{\right|}
\newcommand{\op}[1]{\mathbf{#1}}$
$$\begin{align}
 && \op{F}\_i &= \op{H}^\mathrm{c} + \sum\_j (\op{J}\_j - \op{K}\_j),\\
\text{with}&& 
 \op{J}\_j\lb \phi\_i\rr &=
 \ll \phi\_j(\op{x}\_1) \rb r\_{12}^{-1} 
 \lb \phi\_j(\op{x}\_1) \rr \lb \phi\_i(\op{x}\_2) \rr,\\
\text{and}&&
 \op{K}\_j\lb \phi\_i\rr &=
 \ll \phi\_j(\op{x}\_1) \rb r\_{12}^{-1} 
 \lb \phi\_i(\op{x}\_1) \rr \lb \phi\_j(\op{x}\_2) \rr.
\end{align}$$


While in the Coulomb operator case the $\lb \phi\_i\rr$ doesn't do anything, so you can take it through the proof as a constant:\*
\begin{align}
 \sum\_j\op{J}\_j\lb \phi\_i\rr 
 &= \sum\_j 
 \ll \phi\_j(\op{x}\_1) \rb r\_{12}^{-1} 
 \lb \phi\_j(\op{x}\_1) \rr \lb \phi\_i(\op{x}\_2) \rr\\
 &= \sum\_k \sum\_l \sum\_j U\_{kj}^\*U\_{lj}
 \ll \phi\_k(\op{x}\_1) \rb r\_{12}^{-1} 
 \lb \phi\_l(\op{x}\_1) \rr \lb \phi\_i(\op{x}\_2) \rr\\
 &= \sum\_k \sum\_l \delta\_{kl}
 \ll \phi\_k(\op{x}\_1) \rb r\_{12}^{-1} 
 \lb \phi\_l(\op{x}\_1) \rr \lb \phi\_i(\op{x}\_2) \rr\\
 &= \sum\_k 
 \ll \phi\_k(\op{x}\_1) \rb r\_{12}^{-1} 
 \lb \phi\_k(\op{x}\_1) \rr \lb \phi\_i(\op{x}\_2) \rr\\
 &= \sum\_k \op{J}\_k \lb \phi\_i\rr
\end{align}


That is not the case in the Exchange operator, as it switches the orbitals:
\begin{align}
 \sum\_j\op{K}\_j\lb \phi\_i\rr 
 &= \sum\_j 
 \ll \phi\_j(\op{x}\_1) \rb r\_{12}^{-1} 
 \lb \phi\_i(\op{x}\_1) \rr \lb \phi\_j(\op{x}\_2) \rr\\
 &= \sum\_k \sum\_l \sum\_j U\_{kj}^\*U\_{lj}
 \ll \phi\_k(\op{x}\_1) \rb r\_{12}^{-1} 
 \lb \phi\_i(\op{x}\_1) \rr \lb \phi\_l(\op{x}\_2) \rr\\
 &= \sum\_k \sum\_l \delta\_{kl}
 \ll \phi\_k(\op{x}\_1) \rb r\_{12}^{-1} 
 \lb \phi\_i(\op{x}\_1) \rr \lb \phi\_l(\op{x}\_2) \rr\\
 &= \sum\_k 
 \ll \phi\_k(\op{x}\_1) \rb r\_{12}^{-1} 
 \lb \phi\_i(\op{x}\_1) \rr \lb \phi\_k(\op{x}\_2) \rr\\
 &= \sum\_k \op{K}\_k \lb \phi\_i\rr
\end{align}




---


\* Let 
$$
\mathbb{U}=\left(\begin{matrix}
 U\_{11} & U\_{12} & \dots \\
 U\_{21} & U\_{22} & \dots \\
 \vdots & \vdots & \ddots\\
 \end{matrix}\right);
\mathbb{U}^\dagger\mathbb{U}=\mathbb{E}
\implies U\_{ij}^\* U\_{kl}= \delta\_{ij} = \begin{cases}
 1; & i = j = k = l \\ 0; & i \neq j \dots\\ \end{cases}
.$$


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