Welcome!

My first question would be:  WHEN?

My second question would be:  Where do you go from Kenmare?

Some options for your afternoon / evening in Cork city:  http://www.cork-guide.ie/cork_city/corkcityattractions.html   The English Market is open until 6 PM on Saturday.  If you depart Dublin early enough, say, 10 AM and spend 2 hours at the Rock of Cashel, you could be IN Cork by 3:30 - 4 PM.

Driving from Cork to Kenmare along the N71 can be an all day affair.  I'm not sure you'll have time for both Mizen AND Sheep's Head AND a detour to Baltimore.  A lot will depend upon time of year and the weather.

If you decide to visit Dingle on your full day, it makes no sense to drive back to Kenmare and you would be better off spending that night IN Dingle.

Ok, then -- That's good to know.  

In June, you'll have daylight until 10 PM, provided it's not rainy and cloudy . . .  

Since you'll be passing Cashel in both directions, you could skip it on the way down and drop by on the way back to Dublin.

If you head straight to Cork (again, leaving about 10 AM), you would arrive in Cork about 2 PM.  That would give you plenty of time to visit the English Market.  In that case, staying at the River Lee makes very good sense.  It's location on Western Road is fairly convenient to central Cork city.  Be sure to plot out the route in advance, though, as Cork traffic can be difficult due to the one-way streets and bridges.

I believe the nearest Car Park is:   http://www.q-park.ie/  on Grand Parade, about a block or so from the Market.   http://www.englishmarket.ie/information/locationlayout/

While driving through West Cork you could stop off at Drombeg Stone Circle if you are interested in ancient sites.

On the Beara Peninsula see my article Tea at Josie's. 

I would skip Dingle on this trip unless you are prepared to do two nights in Dublin and sacrifice a night to Dingle.