Are there any known asteroids with average density similar to that of Earth's? In this answer I mention that for very low orbits around spherical bodies, the period tends to scale only as the inverse square root of the density, and not the diameter.

For a low orbits where the semi-major axis is close to the radius of the central body, the period is related to the average density of the body and unrelated to it's size.
So a low orbit around a spherical asteroid (which there usually aren't) made of a mixture of rock and iron (which there usually aren't) will be roughly 90 minutes just like LEO (low Earth orbit), even if it's only 1 km in diameter.

I'd like to use an example, but I don't know where to find an asteroid density table, nor even a histogram, that includes something close to 5 g/cm^3.
Question: Are there any known asteroids with average density similar to that of Earth's?
Some example mean densities from Wikipedia (some rocky planets included for good measure):
body      mean density (g/cm^3)

Earth            5.5
Mercury          5.4
Venus            5.2

Mars             3.9
Io               3.5
Vesta            3.4
Moon             3.3
Hygiea           2.8
Pallas           2.7
Ceres            2.1
Apparently 21 Lutetia (~80x120 km) is one of the most dense asteroids known at about 3.4 g/cm$^3$. Unfortunately that is barely over 60% of Earth's density. 21 Lutetia is unique in that scientists believe that it has a primordial core, that is, it has a core from the formation of the solar system (and it hasn't been destroyed after all these years). We believe it is hidden under the surface. It would be rich in metals, explaining the density, and because it's hidden under the surface, 21 Lutetia would look no different from any other asteroid, which is the case. This asteroid is a prime target for mining because of its metal core. I suppose the answer to your question would be no, unless 3.4 is sufficient for your definition of "close".