I see you have also posted a similar question on the England forum but I think for the short time you have North Wales fits the bill.
It might cost more to fly in Manchester but accommodation and general living expenses will be cheaper in the north than the south of the country.
North Wales has lots of castles in a relatively small area Conwy, Beaumaris, Caernavon, Criccieth, Flint etc. The scenery of Snowdonia is spectacular including a number of waterfalls. For gardens you have Bodnant and Portmeirion.
If you fly into Manchester, I would take the train to Chester for your first night. Its a very pretty, walkable Roman/Medieval city with lots of history. It's got quite a good food scene these days with a few restaurants with very good reputations. As animal lovers if you don't mind a zoo, Chester Zoo is world renowned. It would be a good place to get over jet lag. You could then pick up a car in Chester or catch the train into North Wales and get a car at Llandudno junction.
On your other post you say you plan to travel in August/September. September will be slightly quieter than August but its still peak season and busy. Avoid the last week of August because the last Monday is a public holiday and always busy.