= Solution
There are $L$ primitive cells and therefore $L$ <lattice points>. The specified reciprocal-space density $LA_c/(2\pi)^2$ counts distinct allowed wavevectors; each supports two electron <spin> states. Filling the disk at zero temperature gives
$$
N_e=2\frac{LA_c}{(2\pi)^2}\,\pi k_F^2.
$$
Using $A_c=2a^2$ and $k_F^2=\pi^2/(2a^2)$,
$$
\boxed{\frac{N_e}{L}=\frac{A_c k_F^2}{2\pi}
=\frac{\pi}{2}\simeq1.571\text{ electrons per lattice point}}.
$$
The circle fills a fraction $\pi/4$ of the first-zone area, while one spin-degenerate band has two states per primitive cell; these give the same result. A single-spin count would instead be $\pi/4$, omitting the physical twofold <spin degeneracy> of this model. The circle has only just reached the sides, so the zone corners remain unfilled: first contact does not mean filling the entire first band.
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