  The infinite-layer structure nickelate Ba$_2$NiO$_2$(AgSe)$_2$ (BNOAS) with
$d^8$ Ni ions and a peculiar susceptibility $\chi(T)$ is studied with
correlated density functional methods. The overriding feature of the
calculations is violation of Hund's rule coupled with complete but
unconventional spin-orbital polarization, leading to an unexpected low spin
$^1B_1$, "off-diagonal singlet" (ODS) textured by an internal orbital structure
of compensating $d_{x^2-y^2}^{\uparrow}$ and $d_{z^2}^{\downarrow}$ spins. This
unconventional configuration has lower energy than conventional high-spin or
low-spin alternatives. An electronic transition is obtained at a critical Ni-O
separation $d_c^{Ni-O}=$2.03 \AA, above which Ni becomes magnetic in square
planar NiO$_2$ compounds. We propose scenarios for the signature of magnetic
reconstruction in $\chi(T)$ at $T_{m}$=130 K without any Curie-Weiss background
(no moment) that invoke ordering of Ni $d^8$ moieties that are largely this
generalized Kondo singlet. The underlying physics of this system is modeled by
a Kondo sieve model (2D Kondo necklace) of a "Kondo" $d_{z^2}$ spin on each
site, coupled to a $d_{x^2-y^2}$ spin that is itself strongly coupled to
neighboring like-spins within the layer. The observed magnetic order places
BNOAS below the quantum critical point of the Kondo sieve model, providing a
realization of the previously unreported long-range ordered near-singlet weak
antiferromagnetic phase. We propose electron doping experiments that would
drive the system toward a $d^{9-\delta}$ configuration and possible
superconductivity with similarity to the recently reported Ba$_2$CuO$_{3.2}$
that superconducts at 73 K.
