According to Stoke's law, the retarding force acting on a body falling in a viscous medium is given by $$F=kηrv$$ where $k=6π$.
As far as I know, the $6π$ factor is determined experimentally. In that case, how is writing exactly $6π$ correct since we obviously cannot experimentally determine the value of the constant with infinite precision?
It is not determined experimentally, it is an analytical result. It is verified experimentally.
As @Mick described it is possible to derive the velocity and pressure field of a flow around a sphere in the Stokes flow limit for small Reynolds numbers from the Navier-Stokes equations if the flow is further assumed to be incompressible and irrotational.
Once the flow field is determined, the stress at the surface of the sphere can be evaluated: $$\left.\boldsymbol{\sigma}\right|_w = \left[p\boldsymbol{I}-\mu\boldsymbol{\nabla}\boldsymbol{v}\right]_w$$ from which follows the drag force as: $$\left.\boldsymbol{F}\right|_w = \int_\boldsymbol{A}\left.\boldsymbol{\sigma}\right|_w\cdot d\boldsymbol{A}$$
From this it follows that the normal contribution of the drag force (form drag) is $2\pi\mu R u_\infty$, while the tangential contribution (friction drag) of the drag force is $4\pi\mu R u_\infty$, where $u_\infty$ is the free-stream velocity measured far from the sphere. The combined effect of these contributions is evaluated as $6\pi\mu R u_\infty$ or the total drag force.
This result is also found by evaluating the kinetic force by equating the rate of doing work on the sphere (force times velocity) to the rate of viscous dissipation within the fluid. This shows nicely there are often many roads to the same answer in science and engineering.
For details i suggest you look at the Chapter 2.6 and 4.2 from Transport Phenomena by Bird, Steward & Lightfoot.
If you have read that the 6π coefficient is determined experimentally, then you would also have read that this applies to spherical objects with very small Reynolds numbers in a viscous fluid - Stokes' law is derived by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.
We cannot determine the value of any constant with infinite precision but we can often determine them to a level of precision where the effect of the uncertainty becomes negligible.
I would like to suggest this is not a physics question so much as a history question. I would love to know the exact answer (about the 6) and my historical knowledge is very limited. However this empirical Law defines the viscosity of suitable fluids. For Newtonian (I wish we called call it Stokesic + compare with "Ohmic") fluids the viscosity is defined as 1/(6pi) X the constant of proportionality. This was a choice in defining viscosity and the 6 looks arbitrary but the pi (as elsewhere, with Coulombs Law, which in vacuum is a proper Law about how the universe tics rather than just defining a property of some materials) was included by choice. As a result pi doesn't appear elsewhere in expressing viscosity in terms of the quantities measured in a ball falling determination. Why didn't the clever egg head who defined viscosity of a fluid as 1/(6pi) the constant rather have a Stokes' equation with 4pi/3 ie: D = (4pi/3 )etaxrv
