The KO index is fairly good in warm and muggy situations like last Thursday where convective instability was present (we needed someting to lift the air to get convection started). It takes into account the difference in temperature and moisture between low and mid levels. A KO value of <2 means thunderstorms are likely, as long as there is some lifting mechanism present (from an upper shortwave and/or uplift over terrain/convergence). The sounding from that Wicklow storm below shows a KO Index of 0 (box on the right), which tallies with what happened.
CAPE is the first thing to look at, though. If there is no CAPE then that's it. If there is CAPE then it must be released, which is what didn't happen Thursday afternoon because there was too much of a cap (warm layer of air 1-2 km up) that stopped surface air parcels from rising high enough (to the level of free convection). If the CAPE is released then the Lifted Index (LI) is the next thing to look at as it tells us how many degrees warmer the rising air parcel is than its surrounding air at the 500-hPa level, however this only relates to surface-based convection. A similar index for eleveated convection is the Showalter Index, which is like the LI but for a parcel raised from the 850-hPa level, not the surface. In the sounding above the SI is about -3. A value of -4 or less means severe thunderstorms.
The other indices in that sounding are the Vertical Totals (VT) and Total Totals (TT). The VT is just the difference in temperature between 850 and 500 hPa (the higher the better) and the TT is the VT plus the the difference between the 850-hPa dewpoint and the 500-hPa temperature. A value of >50 means severe thunderstorms are possible and >56 means severe storms with tornadoes.
The EUMeTrain website is an excellent learning tool for lots of different situations. This page describes some of these indices in more detail.