  5$d$ transition metal oxides host a variety of exotic phases due to the
comparable strength of Coulomb repulsion and spin-orbit coupling. Herein, by
pursuing density-functional studies on a delta-doped quasi-two-dimensional
iridate Sr$_3$Ir$_2$O$_7$, where a single SrO layer is replaced by LaO layer,
we predict the formation of a spin-orbital entangled two-dimensional electron
gas (2DEG) which is sharply confined on two IrO$_2$ layers close to the LaO
layer. In this bilayer crystal structure, an existing potential well is further
augmented with the inclusion of positively charged LaO layer which results in
confining the extra valence electron made available by the La$^{3+}$ ion. The
confined electron is bound along crystal $a$ direction and is highly mobile in
the $bc$ plane. From the band structure point of view, now the existing
half-filled $J_{eff}$ = 1/2 states are further electron doped to destroy the
antiferromagnetic Mott insulating state of IrO$_2$ layers near to the
delta-doped layer. This leads to partially occupied Ir upper-Hubbard subbands
which host the spin-orbital entangled 2DEG. The IrO$_2$ layers far away from
the interface remain insulating and preserve the collinear G-type magnetic
ordering of pristine Sr$_3$Ir$_2$O$_7$. The conductivity tensors calculated
using semi-classical Boltzmann theory at room temperature reveal that the 2DEG
exhibits large electrical conductivity of the order of 10$^{19}$.
