In fact in our more "mundane" universe, in 1887, this crucial experiment was performed exactly as I have described by Michelson and Edward Morley in an attempt to detect the existence of the ether. This experiment, named the Michelson-Morley experiment in honor of its authors, however shocked the scientific community by yielding results which strongly implied the non-existence of ether. This result was later on used by Einstein to refute the existence of the ether and allowed him to develop special relativity without this artificial (and sadly for the Steampunk universe non-existent) constraint.
Although Michelson and Morley were expecting measuring different speeds of light in each direction, they found no indication of a different speed in any orientation or at any position of the Earth in its annual orbit around the Sun.
In 1895, Hendrik Antoon Lorentz (1853-1928) concluded that the "null" result obtained by Michelson and Morley was caused by a effect of contraction made by the ether on their apparatus and introduced the length contraction equation
where L is the contracted length, is the rest length, v is the velocity of the frame of reference, and c is the speed of light. Although the main interpretation of Lorentz for this equation was rejected later, the equation is still correct and was the first of a sequence of new equations developed by Poincaré, Lorentz, and others, resulting in a new branch of physics ultimately brought to fruition by Albert Einstein in special relativity. Einstein's idea of space-time contraction replaced Lorentz's interpretation of the contraction equation, and once and for all relegated ether to the history books.