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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 "Epoch of Reionization (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 super(-1) and z = 9.5.