  Kepler's immense photometric precision to date was maintained through
satellite stability and precise pointing. In this white paper, we argue that
image modeling--fitting the Kepler-downlinked raw pixel data--can vastly
improve the precision of Kepler in pointing-degraded two-wheel mode. We argue
that a non-trivial modeling effort may permit continuance of photometry at
10-ppm-level precision. We demonstrate some baby steps towards precise models
in both data-driven (flexible) and physics-driven (interpretably parameterized)
modes. We demonstrate that the expected drift or jitter in positions in the
two-weel era will help with constraining calibration parameters. In particular,
we show that we can infer the device flat-field at higher than pixel
resolution; that is, we can infer pixel-to-pixel variations in intra-pixel
sensitivity. These results are relevant to almost any scientific goal for the
repurposed mission; image modeling ought to be a part of any two-wheel
repurpose for the satellite. We make other recommendations for Kepler
operations, but fundamentally advocate that the project stick with its core
mission of finding and characterizing Earth analogs. [abridged]
