Do you have your own car, or are you hiring one in Ireland and de-hiring in the UK? Cos that would incur penalty fees + carrying a car on the ferry is pricy. Also, it's not really convenient for your needs; it would be better to dehire the car at an Irish airport and rehire one on the UK, because it would save a shed load of boring driving to get to the next area of interest. Dingle Peninsula is one heck of a long drive away from Leeds Castle.
What I would do; fly Cork to Gatwick hire a car, hoof it round to Leeds castle (which you've obviously gathered is nowhere near Leeds).
Go from Leeds castle to Bodian castle to Arundel Castle to Petworth house (no n in Petworth btw) to Guildford. Allow 3 days for that lot. Spend a day in the Cotswolds (Stow on the Wold is great and has a famous bookfair every year) before spending another day in Stratford Upon Avon before leaving in the evening and hoofing it up to the Peak District. Do that in day 6, and then dehire the car in Manchester airport and fly home from there.
This particular itinerary skips Warwick (cos you'll have done several castles by then; for the castle newbie it's quite similar I reckon to Leeds, unless you're REALLY into your castles and can spot the diff's). It also skips Rye (bit of a diversion) and sadly the Lake district (big diversion). You simply have too much to do in too little time, but that's a common problem.
For this itinerary, you definitely need a sat nav and until you've got as far as Guildford I'd leave it in "shortest distance" rather than "fastest route", otherwise it's gonna continually shove you up to the M25, which is definitely not the shortest distance and is often not the fastest route either.
If the Lake District is a must do, option 2, which is also less punishing and gives you more time to actually see stuff instead of lots of roads:-
Fly Cork to Newcastle with Jet 2, hire a car, drive across to the Lakes, specifically Grasmere. Try to avoid it during summer weekends when it gets rammed beyond belief. 1.5 days chalked up.
After that, drop down to the Peak District, do that for 2 more days, then to Warwick Castle and Stratford Upon Avon (which are very close to one another). 1.5 days chalked up.
From there, head on down to Stow on the Wold and Bath, do those for the remaining time, then trundle along the M4 to Heathrow, hand in the car, fly home.
But that would skip Jane Austen country.
Alternatively, you could do the whole English itinerary if you completely skip Ireland as a whole - that buys you 4 more days which then makes the above list fully achievable.
