We knew what to look for the whole time. The mass was unknown, but that simply means you look at all possible mass values.
The 2011 dataset was sufficient to get a hint of a possible new particle at 125 GeV already, but more collisions (from 2012) were needed to clearly confirm that it is a new particle.
At the time of the Higgs discovery, ATLAS and CMS each had about 150-200 Higgs to two photon decays and ~5 Higgs to four lepton decays in the dataset. For individual events, you can never be sure what caused them, finding the Higgs and distinguishing it from background requires a statistical analysis.How much data you need depends on the signal strength, the detector performance, the amount of background, and a bit of randomness. The experiments updated their results once in a while with more data, at the time where the significance was close to 5 standard deviations they decided to make a press conference. A few events more or less wouldn't have changed that.
This Insights article discusses some of the concepts