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.