We report measurements of radon daughters in liquid argon within the MicroBooNE time projection chamber (LArTPC). The presence of radon in MicroBooNE's 85 metric tons of active liquid argon bulk is probed with newly developed charge-based low-energy reconstruction tools and analysis techniques to detect correlated $^{214}$Bi-$^{214}$Po radioactive decays. Special datasets taken during periods of active radon doping enable new demonstrations of the calorimetric capabilities of single-phase neutrino LArTPCs for $\beta$ and $\alpha$ particles with electron-equivalent energies ranging from 0.1 to 3.0 MeV. By applying $^{214}$Bi-$^{214}$Po detection algorithms to beam-external physics data recorded over a 46-day period, no statistically significant presence of radon is detected, corresponding to a limit of $