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I would do the following:

  1. Translate the polytope to have one vertex at the origin. This way it lives in a $k$-dimensional vector subspace, not only affine subspace.

  2. Find an orthonormal basis of this subspace.

  3. Express the polytope in that basis.

  4. Now work in $\mathbb{R}^k$ and find the volume of the polytope.