  We present techniques for bridging the gap between idealized inverse
covariance weighted quadratic estimation of 21 cm power spectra and the
real-world challenges presented universally by interferometric observation. By
carefully evaluating various estimators and adapting our techniques for large
but incomplete data sets, we develop a robust power spectrum estimation
framework that preserves the so-called "EoR window" and keeps track of
estimator errors and covariances. We apply our method to observations from the
32-tile prototype of the Murchinson Widefield Array to demonstrate the
importance of a judicious analysis technique. Lastly, we apply our method to
investigate the dependence of the clean EoR window on frequency--especially the
frequency dependence of the so-called "wedge" feature--and establish upper
limits on the power spectrum from z = 6.2 to z = 11.7. Our lowest limit is
Delta(k) < 0.3 Kelvin at 95% confidence at a comoving scale k = 0.046 Mpc^-1
and z = 9.5.
