If you have 10 days, you should travel around Ireland. Consider flying Ryan Air to Dublin and then flying back to London from Shannon.

If possible, rent a car when you are ready to head out of Dublin.

My favorite part of Ireland is in the west: Galway and County Mayo.

Dublin is a lovely city...but it is a city. I'd suggest only 2 nights there. Stop at an Irish Tourist Board and book a B&B near Galway. It's about 4 hrs.+ driving. Eat in the Old Quay area. Then head up into Co. Mayo. You can go either along Lough Corrib, or the other way which is out via Salt Hill area.
You could spend a couple of nights in Co. Mayo making a loop that will bring you back south to the Galway area.

At that point, continue south along the coast...see The Cliffs of Moher...Doolin...maybe take the car ferry at Tarbet...and go to Killarney.

You can drive the Ring of Kerry.

When you're ready to head back to London, leave from Shannon.

Karen