6 nights and 5 full days in Ireland

Given the limited time you have available, coupled with the limited mobility, you're probably best advised to stick close to Dublin.
There's plenty to do in the city itself to keep you occupied for 5 days, and possible day trips would be to the Glendalough monastic settlement, Powerscourt, Newgrange and the Boyne Valley. Could also take the train up to Belfast - it's a fascinating city and would be well worth a day or two of your time. Check the Trip Advisor "Things to do" feature for attractions and activities that most appeal to you in and around Dublin. That site also lists a number of tour companies that may be of interest. One could probably craft a customized tour for the 2 of you if there's any interest in doing that.
Though Galway and the west of Ireland are beautiful I think they're probably best saved for a future visit when you have more time to dedicate to exploration. Just getting there and back via public transportation will eat up most of your available time. Better to do it with a car on a future trip.

If you want to see the Cliffs of Moher (which I highly recommend) then I'd probably skip Galway. It's not a bad drive from Dublin but you would definitely need to spend a night or two on the west coast so you're not wasting all your time driving back and forth. I'd recommend getting to the cliffs early in the morning or later in the afternoon after all the tour buses have left.

Greetings from Ireland,

If you do have your heart set on visiting either Galway or Cork, its perfectly doable. Remember as Rick always reminds people, Ireland is roughly the same size as West Virginia!

You can take a train from central Dublin to Central Galway, about 3 hours duration and very comfortable, buy the tickets online at www.irishrail.ie
Then take a day trip from Galway to the cliffs of Moher with 'Galway Tour Company' or to the Aran Islands with 'Lallys'

Same for Cork, take the train and then a day trip to taxi out to Blarney Castle, its only 6 miles from the city.

Id go for Galway though, smaller and more compact and more to do in terms of day-trips. Theres also Connemara.

Slainte/Good health
Stephen McPhilemy
Rick Steves Ireland Tour-Guide

I agree with Stephen - seeing at least a part of the west coast is perfectly doable. I took my son to Ireland years ago, and when we arrived on the west coast, with its rolling hills and little stone fences, he said," Now I feel like I'm in Ireland!" Dublin is great, but it's a city. You won't get the feeling of being in Ireland if that's all you see.