Kepler's laws hold only for bodies orbiting what can be approximated as point masses. On scales as large as planetary motion, even stars can be approximated as point masses. Galaxies, however, can not, and so Kepler's laws don't hold.
In fact, there isn't a uniform density distribution of dark matter; generally, data is fitted using a Navarro-Frenk-White density profile1,2:
$$\rho(r)=\frac{\rho_0}{\frac{r}{R_s}\left(1+\frac{r}{R_s}\right)^2}$$
for some density parameter $\rho_0$ (not the central density) and a scale length $R_s$; this works well in most areas, although it fails at the galactic center, where $\rho\to\infty$ as $r\to0$. Also, note that this is the wrong density distribution to yield Keplerian behavior. Furthermore, this profile shows that for most $r\ll R_s$, $\rho(r)\sim r^{-1}$, and in all cases, the density decreases as you get further out from the center. The dark matter halo is not uniformly distributed.
Regarding your statement about constant velocity, I'd recommend looking at some rotation curves extrapolated for many galaxies. It's true that after a certain radius, the curve seems to be relatively flat, but there are actually plenty of irregularities and oscillations, and in some cases, the velocities even tail off a little at the outer reaches of the galaxy. There's enough variation - and certainly not an artefact of experimental error - to cast aside any doubts that there's a giant coincidence here; I challenge you to find a curve which is perfectly flat after a certain peak.
You also may be interested in answers by Kyle Oman and Rob Jeffries.
- For more information, start reading the original papers, e.g. Navarro et al. (1996).
- The Einasto profile is another popular choice; it uses an exponentially decreasing radial density model with finite central density.
