The husband of Deirdre Morley is asking the HSE Mental Health Services for an investigation into her diagnosis, treatment and medication prior to the tragedy, as a matter of urgency.
In a statement, Andrew McGinley says today's verdict is "probably the right one" and that everyone who knows Deirdre knows how much she loved their children Conor, Darragh and Carla and how devoted she was to them.
He added he's no closer to understanding why his children died, but he will "continue to celebrate their all too short lives".
It comes after Ms Morley was found not guilty of murdering her children by reason of insanity.
The 44-year-old nurse was found to be suffering from a severe mental illness when she killed the children last year.
In a statement after the verdict, Mr McGinley said: "Today’s verdict is probably the right verdict. Everyone who knows Deirdre, knows how much she loved our children and how devoted she was to them.
"Whatever the outcome of this trial, it remains that our beloved children Conor, Darragh and Carla have died. As I write this, I’m no closer to understanding why."
He said the trial was never going to explore Deirdre's diagnosis, treatment and medication prior to the tragedy, so the family now wants an "inclusive investigation" by the HSE Mental Health Service.
He said: "We ask for this to be conducted as a matter of urgency. This will help us understand the insanity that took the lives of our beloved Conor, Darragh and Carla.
"We as a family need to be included in any investigation as our exclusion during her treatment has left us with many unanswered questions. We believe that an inclusive investigation can only serve to inform clinicians in their practice and therefore avoid tragedies like ours happening again. We do not want any other family to suffer as we have.
"It is too late for us but I do not want to see another grieving parent speaking in the future about the same exclusion after a similar catastrophic loss. My message here and now to anyone who has a loved one in psychiatric care is to get in there as soon as you can to be added as an advocate for their treatment plan."
Mr McGinley said he will continue to celebrate the "all too short lives" of his three children so they are never forgotten.
He said: "Conor had asked me to help him with a YouTube channel and I promised I would. You will find Conor’s Clips on YouTube now and given his interest in the internet this is supported on Twitter and Instagram. I hope to continue to do that with the same great sense of humour which Conor had.
"I promised to build a snowman for Carla. As we don’t get snow every year I’m planning a colouring competition called Snowman for Carla in her name.
"For Darragh I promised to get involved with Rathcoole Boys. However Darragh had been involved in everything in the community – Rathcoole Boys, Commercials Hurling, St Marys GAA, The Athletics Club, the Drama Club to name but a few. In Darragh’s name I am starting a charity to encourage everyone to get involved more in their own communities and to participate … As Darragh Did.
"Personally, I will be taking some of the unique characters that Conor and Darragh created in their own comics, plays and books and I will attempt to do them proud by bringing these characters further along in books which I hope to write in their honour."