In your
\[Tau]z\[Theta]2[\[Xi]_, z_] :=
definition you have a space missing between the \[Xi] and the D on the right hand side.
In your
\[Tau]zr2[\[Xi]_, z_] :=
definition you have (\[Eta]^2)[z] which should probably be \[Eta]2[z] on the right hand side.
With those changes Solve seems to return a result in a few seconds.