Kate, what a wonderful idea for your mother's 60th birthday! You'll get plenty of good advice here.

Since you only have 4 days, my recommendation is to head in the direction of one of your chosen sites, and stay there.

If you choose the Cliffs of Moher, you can fly into Shannon airport, head up to County Clare, and stay in one of the little villages near the Cliffs of Moher. (Ballyvaughan is one of them for example.) From there, you can enjoy the Cliffs of Moher, the Burren nature area, and the Dunguaire castle medieval banquet, plus a number of fun musical pubs in the various villages.

OR choose to head for Dingle. Stay in Dingle, enjoy the town, do the Dingle peninsula, maybe take a boat out to the Blasket islands, and from there, if you had enough time, there are other day-tour options.

OR head towards the Ring of Kerry area, and see the sites in that area.

Another idea is to choose either the Ring of Kerry OR the Dingle peninsula, as you don't really need to see both in such a short visit. Then try to add the Cliffs of Moher.

Have a wonderful time! Michele's book is great, by the way...fulll of lovely places to stay, sound advice, and tasty places to eat.