There is a proof by the author of a stronger inequality.    
First follow the argument at the top of page 486 of <A HREF="https://cedricvillani.org/sites/dev/files/old_images/2012/08/P09.ICM_1.pdf">Hypocoercive diffusion operators</A> to arrive at    
<IMG SRC="https://ilorentz.org/beenakker/MO/Villani_1.png" WIDTH="500"/>    
Then follow the proof in <A HREF="https://arxiv.org/abs/math/0609050">arXiv:math/0609050</A>, page 152, where it is shown that the functional
<IMG SRC="https://ilorentz.org/beenakker/MO/Villani_2.png" WIDTH="500"/>    
evolves in time such that
<IMG SRC="https://ilorentz.org/beenakker/MO/Villani_3.png" WIDTH="500"/> 

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Alternatively, for the case $n=1$, $V=0$ there is a worked-out proof in page 10 and following of <A HREF="https://hal.archives-ouvertes.fr/hal-01616979/document">Héraou's lecture notes.</A>    
Let me summarize the key steps. As a short-hand notation we write $||\partial_x h||^2=\int (\partial h/\partial x)^2\,d\mu$ and $\langle\partial_x h,\partial_v h\rangle=\int (\partial h/\partial x)(\partial h/\partial v)\,d\mu$. We will make use of the identities
$$\langle f,\partial_v g\rangle=\langle(-\partial_v+v)f,g\rangle,$$
$$\langle v\partial_x f,f\rangle=0$$
$$[\partial_v v,\partial_x]=1,$$
and the Cauchy-Schwartz inequality
$$2|c\langle\partial_v f,\partial_x f\rangle|\leq c^2||\partial_v f||^2+||\partial_x f||^2.$$
The Fokker-Planck equation for $n=1$, $V=0$ reads
$$\partial_t h+v\partial_x h+(-\partial_v+v)h=0.$$
This implies the derivatives
$$-\frac{1}{2}\frac{d}{dt}||h||^2=||\partial_v h||^2,$$
$$-\frac{1}{2}\frac{d}{dt}||\partial_x h||^2=||\partial_v\partial_x h||^2,$$
$$-\frac{1}{2}\frac{d}{dt}||\partial_v h||^2=\langle\partial_x h,\partial_v h\rangle+||(-\partial_v +v)\partial_v h||^2=\langle\partial_x h,\partial_v h\rangle+||\partial_v^2 h||^2+||\partial_v h||^2,$$
see page 10 of the cited lecture notes.

So the derivative $dI/dt$ in the OP reduces to
$$\frac{dI}{dt}=-2a||\partial_v\partial_x h||^2-2c\biggl(\langle\partial_x h,\partial_v h\rangle+||\partial_v^2 h||^2+||\partial_v h||^2\biggr)$$
$$\qquad\leq -2a||\partial_v\partial_x h||^2-2c||\partial_v^2 h||^2-(2c-c^2)||\partial_v h||^2+||\partial_x h||^2.$$
It remains to bound $||\partial_x h||^2$. In the lecture notes they do this by adding the mixed term $b\langle\partial_x h,\partial_v h\rangle$ to the left-hand-side of the inequality, which then gives a term $-2b||\partial_x h||^2$ on the right-hand-side to dominate.