Not sure how much of this you have already got, but did some digging (in lieu of more pressing things such as house work!)
Harriot Fernell baptised 4 Oct 1786, Chesterfield. The baptism register gives her father as ‘Mr John Burgoyne Fernell’ - so a 'Gentleman’.
She married Thomas Finney 12th Oct 1807 at Chesterfield, by licence - she signs Harriet Furnell. The marriage entry (difficult to find as not images for Chesterfield on any of the usual sites - had to go to familysearch catalogue and go through the actual film), tells us she can sign her name. A withess is Ann Burgoyne Needham - no doubt a married sister (she crops up later in mother Betty’s will). Also I think, her brother W(illiam) B(urgoyne) Fernell is a witness.
The licence tells us she was aged 21 or over.
Also with Thomas Finney, gentleman of Hartington, ‘William Burgoyne Fernell, gentleman’, goes to swear the licence. WB Fernell must have one of the most flourishing signatures I’ve ever seen!
The will of John Burgoyne Fernell in 1789 says he is of ‘Spring House’, within the Liberty of Calow, parish of Chesterfield, and a Gentleman.
Also has a property in Southwark, Surrey. (in Will Street? - bit difficult to make out).
Also a property called School End in Oakingham, Berkshire.
And ‘several closes, Lands hereditaments and premises’ in Calow in the possession of Jethro Adlington.
John Burgoyne Fernell married Betty Adlington 13th Jan 1773 at Chesterfield.
The will of his wife Betty Furnell in 1798 also gives her as of ‘Spring House’.
Some interesting snippets in this. She mentions her husband’s ‘gold topped cane’ and watch. Also’ fowling piece and brace of pistols’. ‘Four shooting pieces in the Hall’, and the Painted Arms of the Furnell (sic) Family. Also ‘a painting of shipping over the sideboard’. Perhaps he had been to sea at some point? Wild speculation on my part!
John Bourgoyne Fernell was baptised 16th July 1750 at St Mary Magdelene Bermonsey, Surrey. Also a brother William baptised there, both of William Fernell and Ann.
PR gives father William Fernell as ‘Gent’.
So John didn’t have much time to ‘go to sea’ - certainly not to get to the rank of Captain anyway - he was born 1750, and married 1773, so only 23 when he got married. And died 1789, so only 39 years old.
William Fernell (John's father) married Ann Burgoyne 8th Dec 1748 at Rotherhide by licence. He is of St Marys Rotherhide, and a widower. She is of the same parish, spinster.
William, John ‘s brother, writes a will in 1772.
He is of the parish of St Mary Magdelene. Bermonsey, ‘Gentleman’, and is going off ‘on a voyage’ to be ‘fifth mate’ on ‘the good ship Clive’ under Captain John Allen to the East Indes. (Fifth mate sounds pretty lowly to me for a ‘gentleman’?).
Leaves mostly to his brother John Burgoyne Fernell (no place of residence given).
Nowhere have I come across ‘Keeler Hall’, or the fact that John Burgoyne Fernell was a ‘Captain’, as in the notes. He seems always to be just given as ‘gentleman’.
I did find this interesting little snippet in the Chesterfield register from 1784.
Perhaps be was called up to be in this - possibly because he was Mr Fernell, he was made a ‘Captain’? Especially considering all those guns he possessed.
(Or like my grandfather - who died when my father was only 2, but was evidently called ‘Captain’. No evidence whatsoever to verify this. He worked on the railways. Perhaps because he sailed to Australia in 1884 to start a new life, but it all turned to custard and he returned to England, he gave himself the self styled epithet ‘Captain’. Who knows?!)
Sorry, not very helpful as to 'Keeler Hall'. Google throws up nothing at all.
I think the notes you have are type-written? In which case it could have been a transcription error somewhere along the line. Just an idea.