Reason for 6π factor in Stokes' law

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.