The universe doesn't expand on such small scales. In fact there is no expansion even on the scale of the local group of galaxies. It is only when you look at more distant galaxies that you see the expansion of space.
The basic assumption of expansion is that the universe is uniform in all directions. This isn't true, not even nearly true when you are inside a cluster of galaxies.
So in theory there should be no expansion on the scale of a km or so.
LIGO is not designed to detect such changes. It is designed to detect mid-frequency gravitational waves, these have a frequency that is measured in Hertz. Cosmological expansion occurs at a time scale that is several orders of magnitude slower, and so wouldn't be detected by LIGO in its current configuration (and it isn't clear to menme that it could be detected at all, due to all the other sources of strain, even if expansion was occurring on a km scale).