"A party was formed this evening for going on the following day to see a very fine place about twelve miles from Barton, (…) The grounds were declared to be highly beautiful, and Sir John, who was particularly warm in their praise, might be allowed to be a tolerable judge, for he had formed parties to visit them, at least, twice every summer for the last ten years. They contained a noble piece of water; a sail on which was to form a great part of the morning's amusement;"

An example of such a house is Ugbrooke, 10 miles from Exeter.
"In an open valley on the other side of this magnificent woody ridge is Ugbrooke, the seat of Lord Clifford, it is placed on a gentle rising of the ground, on the side of a fine piece of water; and trees of every species crown the summits around it, or stand insulated with their gigantic bulk." Williams, T.H. Picturesque excursions in Devonshire: the environs of Exeter 1815/16.

The grounds were laid out by Capability Brown in the mid C18th and the largest of the lakes looks like it could take boating parties. Above is an aerial view and below is a link to the house and grounds.

It seems like they understood the sort of place that Whitwell was and I can appreciate the level of disappointment they must have felt at not going, particularly at such short notice.

It is possible that JA visited Ugbrooke when she stayed in Dawlish: