Gravitationally lensed supernovae (glSNe) are of interest for time delay cosmology and SN physics. However, glSNe detections are rare, owing to the intrinsic rarity of SN explosions, the necessity of alignment with a foreground lens, and the relatively short window of detectability. We present the Las Cumbres Observatory Lensed Supernova Search, LCOLSS, a targeted survey designed for detecting glSNe in known strong lensing systems. Using cadenced $r^\prime$-band imaging, LCOLSS targeted 114 galaxy-galaxy lensing systems with high expected SN rates, based on estimated star formation rates. No plausible glSN was detected by LCOLSS during our two year observing program. We carry out an analysis here to measure a detection efficiency for these observations. We then perform Monte Carlo simulations using the predicted supernova rates to determine the expected number of glSN detections. The results of the simulation suggest an expected number of detections and 68 per cent Poisson confidence intervals, $N_{\mathrm{ SN}} = 0.20, [0,2.1]$, $N_{\mathrm{ Ia}} = 0.08, [0,2.0]$, $N_{\mathrm{ CC}} = 0.12, [0,2.0]$, for all SNe, Type Ia SNe, and core-collapse (CC) SNe, respectively. These results are broadly consistent with the absence of a detection in our survey. Analysis of the survey strategy can provide insights for future efforts to develop targeted glSN discovery programs, especially considering the large anticipated yields of upcoming surveys.