Take your expression:
$$ v=\frac{\omega}{k} \tag{1} $$
We'll write $\omega=2\pi f$, rather than $\omega = 2\pi/\tau$, where $f$ is the frequency of the wave and substitute for $\omega$ and $k$ in equation (1) to get:
$$ v = f\lambda \tag{2} $$
The wavelength $\lambda$ is the distance the wave moves in one cycle, and the frequency is the number of cycles per second, so if we multiply these together we get the distance moved per second. And the distance moved per second is of course just the speed. That's why the equation (2) applies.
So equation (1) and its equivalent form equation (2) aren't telling us anything very exciting. They just tell us there is a relationship between the frequency, wavelength and velocity without telling us anything about which if any of these are constant. As it happens the velocity of sound in an ideal gas is given by:
$$v = \sqrt{\gamma\frac{P}{\rho}} $$
where $P$ is the pressure and $\rho$ is the density of the gas. $\gamma$ is a constant called the adiabatic index. Or with some messing around we can rewrite this in terms of the temperature:
$$ v = \sqrt{\gamma RT} $$
So (for an ideal gas) the speed of sound is determined by the temperature, and it's constant when the temperature is constant.