The driving times are based I believe on distances at an assumed constant speed that unless you are lucky and experienced you can't maintain. On most Irish roads you can't maintain speed because you are pulling over and virtually stopping to let a car pass on a narrow road, or on a narrow two lane road not maintaining speed because you slow down as you pass another car with perhaps a foot to spare......then there is getting stuck behind a tour bus, farm tractor or the herd of sheep. It is a constant delay.
As far as bases go.....it depends on what you most want to see and what your kids like. History, cool scenery, shopping, the family friendly pubs and restaurants, small towns etc ?
For example just outside of Dublin is Wicklow with gorgeous mountain drives, Sallys Gap where Braveheart was filmed, Glendalough with its insanely beautiful lake walks, monastic ruins and round tower, and great small towns. If you based there you could easily fill 3+ days with fun things to see. Head just an hour south and take a boat ride to the Saltee Islands and see incredibly diverse species of birds many of which you can walk right up to.
The Cliffs of Moher are beautiful as is the surrounding area.....touristy but wonderous. But there is lots to see in the area.
Three of four days in the Ring of Kerry won't even begin to plumb its depths with ringforts, castles, famine homes, spectacular scenery, beaches, wonderful small towns like Sneem, Waterville, Ballinskellig etc. On the Skellig Ring you could stop for world famous chocolates and then show the kids either from a distance or on a boat ride Skellig Michael where Star Wars was filmed. What is the best is taking the time to stop, wander, explore and meet folks. Far better than any main tourist stops are the Irish people...genuine, funny, welcoming and so generous.
I've mentioned it before but our best memories are taking the time near Cashel to talk to an elderly lady and her "fine excuse for a dog". Since we didn't have a maxed out schedule we ended up talking to her and getting invited to her home for homemade scones. Or in Ballinskelkig stopping to look at horses and having the farmer come out and talk with us ...ask us to help him feed them and then have tea with him. Or having a fisherman in Portmagee start talking to us as we looked out around the coast and him giving us some of his fresh catch for dinner and hearing his stories of being caught out in a storm. Or even the fun of a relaxed breakfast in Cahairseveen with fresh teabrack and fresh ground hot chocolate with the nicest baker/owner you could ever meet. In Ireland that stuff happens constantly as long as you leave time for it.
You really can't go wrong. Wicklow, Kerry, Dingle, Donegal, Connemarra or others .....all have so much to see and do. Just don't try to do everything....and avoid long drives and one night stays when possible.