I understand that at the middle of the whirl can't be any water because that would make the velocity of water infinite.
But what is the mechanism, the force, the structure that removes water from the center?
For low draining rates you can have water at the centre of the whirl. The free vortex model in which tangential velocity of the whirl goes as $1/r$ (where $r$ is the radial distance from the whirl's centre) is an idealization. In that model you get infinite velocity at the centre ($r=0$) because it is assumes that infinite amount of vorticity is concentrated at the centre of the whirl. But real fluids have viscosity which will not allow such infinite concentration and instead smear the vorticity over a finite radius. A better model for such whirls is a Rankine vortex which contains a core of radius $R$ which is in solid body rotation and therefore in this region the velocity goes as $r$, while outside this core the velocity goes as $1/r$. Thus the singularity is avoided by nature.