The reason is the fact that dark matter is non collisional. The dark matter particles interact only gravitationally, they feel no pressure, right, but they also feel no drag! No drag, no friction, means that they can't dissipate energy.
Imagine a cloud of dark matter that starts very large and diffuse. Initially it has a very small kinetic energy $K$ and small negative potential energy $U$. If you let it evolve, initially gravity will make it collapse, but the virial theorem tells us that it will reach an equilibrium where $U=-2K$. This equation, plus the conservation of energy $U+K=const$ uniquely determines $U$ and $K$, as well as the radius of the cloud, that will be of the order of $R \approx GM^2/U$.
If dark matter could dissipate energy, the total energy would change and the cloud would shrink further, but since it can't, it is locked in viral equilibrium for the time being (or until it evaporates, but this is out of the scope of the question)
Regarding the actual density profile of the dark matter halo, which you suspect has been fitted to the data, the profile is often inferred from N-body simulations of dark matter. One famous example is the Navarro-Frenk-White profile
$$ \rho(r)={\rho_0 \over {r \over R_s} \left(1+ {r\over R_s}\right)^2}$$
So, the shape is determined from simulation, while the two free parameters $\rho_0$ and $R_s$ must be instead fitted, because they vary for each halo. Mind that this is no "cheating", it is akin to fitting a law to the experimental data, which is done in every field of physics.
