Someone should mention an elementary answer the specific question by OP. An affine hyperplane in $\mathbb{R}^n$ can be uniquely represented as $l_{v,c}:=\{x\in\mathbb{R}^n:v\cdot \mathbb{x}=c\}$, where $v\in\mathbb{R}^n$, $|v|=1$, and $c>0$ (except for planes passing through the origin which can be disregarded as they will have zero measure anyway). Let $dv$ be the uniform measure on the unit sphere and $dc$ the Lebesgue measure on the real line. Then, if $\mathcal{L}$ denotes the map $(v,c)\mapsto l_{v,c}$, the measure on the set of planes defined by $\mu(A)=dv\otimes dc(\mathcal{L^{-1}}(A))$ satisfies all the requested properties*. Indeed, for example, the effect of a translation by a vector $-w$ in these coordinates is the map $(v,c)\mapsto (v,c+v\cdot w)$ or $(v,c)\mapsto (-v,-c-v\cdot w)$, both of which have Jacobian equal to one. The effect of rotation is simply rotating $v$, the effect of scaling is to scale $c$ etc.

The same can be done for affine subspaces of all dimensions - an affine subspace is an element of the Grassmanian shifted by a vector in its orthogonal complement, and you can just take the Haar measure on the Grassmanian times the Lebesgue measure on its orthogonal complement.

*except that the scaling property is different, but it should not be hard to see that there are no non-trivial measures with the requested scaling property.

Someone should mention an elementary answer the specific question by OP. An affine hyperplane in $\mathbb{R}^n$ can be uniquely represented as $l_{v,c}:=\{x\in\mathbb{R}^n:v\cdot \mathbb{x}=c\}$, where $v\in\mathbb{R}^n$, $|v|=1$, and $c>0$ (except for planes passing through the origin which can be disregarded as they will have zero measure anyway). Let $dv$ be the uniform measure on the unit sphere and $dc$ the Lebesgue measure on the real line. Then, if $\mathcal{L}$ denotes the map $(v,c)\mapsto l_{v,c}$, the measure on the set of planes defined by $\mu(A)=dv\otimes dc(\mathcal{L^{-1}}(A))$ satisfies all the requested properties*. Indeed, for example, the effect of a translation by a vector $-w$ in these coordinates is the map $(v,c)\mapsto (v,c+v\cdot w)$ or $(v,c)\mapsto (-v,-c-v\cdot w)$, both of which have Jacobian equal to one. The effect of rotation is simply rotating $v$, the effect of scaling is to scale $c$ etc.

The same can be done for affine subspaces of all dimensions - an affine subspace is an element of the Grassmanian shifted by a vector in its orthogonal complement, and you can just take the Haar measure on the Grassmanian times the Lebesgue measure on its orthogonal complement.

*except that the scaling property is different, but it should not be hard to see that there are no non-trivial measures with the requested scaling property.

UPD: To see that such a measure is unique up to scaling, observe that if we have another such measure $\nu$, then $\nu$ is absolutely continuous with respect to $\mu$, thus its pullback to $\{(v,c)\}$ has a density with respect to Lebesgue measure. Clearly, this density is rotationally invariant, thus, it is enough to show that it is invariant under shifts $c\mapsto c+\alpha$. Let $\varepsilon,\delta>0$ be small, let $|v_0|=1, c_0>0$ and consider the set $R_{\varepsilon,\delta} (v_0,c_0)=\{|v-v_0|<\varepsilon,|c-c_0|<\delta\}$. We have $1-\varepsilon^2<v\cdot v_0<1+\varepsilon^2$ for all $v\in R_{\varepsilon,\delta} (v_0,c_0)$. Hence, shifting the corresponding set of lines by $\alpha v_0$ sends $R_{\varepsilon,\delta} (v_0,c_0)$ to a set contained in $R_{\varepsilon,\delta+\varepsilon^2} (v_0,c_0+\alpha)$ and containing $R_{\varepsilon,\delta-\varepsilon^2} (v_0,c_0+\alpha)$. Sending $\varepsilon\to 0$ and using (e. g.) the Lebesgue differentiation theorem completes the proof (of course, one can also do it without densities).