  The Large Magellanic Cloud (LMC) has an extensive H$_{\alpha}$ emission halo
that traces an extended, warm ionized component of its interstellar medium.
Using the Wisconsin H$_{\alpha}$ Mapper (WHAM) telescope, we present the first
kinematic \ha\ survey of an extensive region around the LMC, from $l,b =
(264\deg .5,\,-45\deg .5)$ to $(295\deg .5,\,-19\deg .5)$, covering $+150\leq
v_{lsr} \leq +390~ km s^{-1}$. We find that ionized hydrogen exists throughout
the galaxy and extends several degrees beyond detected neutral hydrogen
emission $(\log{\left(N_{\rm H_{~I}/\rm cm^{-2}}\right)\approx18.3})$ as traced
by 21-cm in current surveys. Using the column density structure of the neutral
gas and stellar line-of-sight depths as a guide, we estimate the upper limit
mass of the ionized component of the LMC to be roughly
$M_\mathrm{ionized}\approx (0.6-1.8)\times 10^{9}\,\mathrm{M}_{sun}$, which is
comparable to the total neutral atomic gas mass in the same region
($M_\mathrm{neutral}\approx0.75-0.85\times10^{9}\,\mathrm{M}_{sun}$).
Considering only the atomic phases, we find
$M_\mathrm{ionized}/M_\mathrm{ionized+neutral}$, to be 46\%--68\% throughout
the LMC and its extended halo. Additionally, we find an ionized gas cloud that
extends off of the LMC at $l,b \approx (285\deg, -28\deg)$ into a region
previously identified as the Leading Arm complex. This gas is moving at a
similar line-of-sight velocity as the LMC and has
$M_\mathrm{ionized}/M_\mathrm{ionized+neutral} =$ 13\%--51\%. This study,
combined with previous studies of the SMC and extended structures of the
Magellanic Clouds, continues to suggest that warm, ionized gas is as massive
and dynamically-important as the neutral gas in the Magellanic System.$
