  We present year-long, near-infrared Hubble Space Telescope WFC3 observations
of Mira variables in the water megamaser host galaxy NGC 4258. Miras are AGB
variables that can be divided into oxygen- (O-) and carbon- (C-) rich
subclasses. Oxygen-rich Miras follow a tight (scatter $\sim 0.14$ mag)
Period-Luminosity Relation (PLR) in the near-infrared and can be used to
measure extragalactic distances. The water megamaser in NGC 4258 gives a
geometric distance to the galaxy accurate to 2.6% that can serve to calibrate
the Mira PLR. We develop criteria for detecting and classifying O-rich Miras
with optical and NIR data as well as NIR data alone. In total, we discover 438
Mira candidates that we classify with high confidence as O-rich. Our most
stringent criteria produce a sample of 139 Mira candidates that we use to
measure a PLR. We use the OGLE-III sample of O-rich Miras in the LMC to obtain
a relative distance modulus, $\mu_{4258} - \mu_{LMC} = 10.95 \pm 0.01 $
(statistical) $\pm 0.06 $ (systematic) mag in good agreement with the relative
distance determined using Cepheids. These results demonstrate the feasibility
of discovering and characterizing Miras using the near-infrared with the Hubble
Space Telescope and the upcoming James Webb Space Telescope and using them to
measure extragalactic distances and determine the Hubble constant.
