This is actually a little complicated and will depend on the type of optical fiber and LED used. There is a big difference light from a laser diode into a fiber and getting light from an LED into fiber. Basically the LED acts as an extended source and can't be efficiently focused into the small area of the fiber. The LED is well collimated optically is much more like a point source and can be focused to a very small spot.  "Etendue" is the technical term. If you look at some optical simulation  websites like Z-Max or optalix or other optical modeling software sites you will find a bunch of ray-tracing diagrams. that may be helpful. In for a large multimode fiber, you need to match the numerical aperture of the lens with the N.A. of the fiber. Even then you may have pretty low efficiency.  

Is there a reason you are making these choices?  I just "googled" LED fiber optic light source" and found lots of stuff.  Using an LED with and fiber optic to illuminate stuff is a rather old problem, with a variety of possibly solutions, and there are many well known, well-engineered solutions.  Seems to me you are re-inventing the wheel.   And, your question seems to me likely not too different from "how well does an octagon roll?"

Try to measure the change in light intensity of light from the source to the point where it interacts with the sample using a light intensity sensor such as a photo diode or a CCD camera. Power loss will be proportion to the square of change in light intensity

What kind of LED? What kind of fibre, Single mode, or Multimode, core size/aceptance angle, etc.??

If the fibre modes are fully and equally excited, the only factor is the equivalent brightness of the LED. So for surface emitting LEDs, lenses may make the launch scenario more convenient (if you cannot get cliose enough), but they do not offer gains in launch power. Superradiant LEDs provide optical power in fewer spatial modes and with higher intensity. Personally I would recommend acquiring an LED with integrated fibre pigtail and published spec for the power exiting the fibre. Probably better not to waste your time and effort on problems that others have already solved for high volume commercial applications (fibre communications).

Would it be possible to simply measure the light at the end of your fiber usign a powermeter (silicon photodiode). Thorlabs sells decent ones at a good price.

If you need more detailed information on the actual profile of your output beam, I recommended a beam profiler (e.g. DataRay).

You can model this and estimate the light you're losing, but this will not only be complex depending on the profile of your LED but also probably inaccurate. If you can measure it, it will be more trustworthy.