"How has she failed as a mother" you may ask? well. first off, her leaving winterfell and all her boys which Eddard has SPECIFICALLY asked her to watch over and went down to kings landing with Rod (who could have gone alone no problem to talkt o eddard, why did caitlyn need to come)
Circumstances intervened here.
Eddard told her to stay in Winterfell, before someone sent a footpad to kill Bran. We're privy to Eddard's POV chapters, and he never thought to himself that Cat had screwed up or disobeyed him by coming to King's Landing.
and then go later on and do the worst she could possibly do by letting the kingslayer go, through miles and miles of outlaw infested terrain with nobody but one bodyguard? all the while hoping that the oh so valliant knight who have no honor(at the time) to go and fufil himself and bring back the darling stark girls who were his enemy.
As Jaime himself thinks, though, Catelyn was trusting in Tyrion's honor, not his (Jaime's).