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Thanks for the mention re Jamaican Family Search - it has been useful, particularly the Malabre Manuscript (Malabre married into the DeGournay family)

The main problem is trying to find the name "King", some documents have Kingston on every entry 

There were a couple of land grants that were of interest
In 1753 there was a grant of 300 (assumed) acres in Manchioneal to Peter King and family (4 in total)
In 1741, 300 acres in Manchioneal to James Bryan and family (7 in total)
In 1742, 300 acres in Manchioneal to Charles Bryan and family (1 in total, so not sure why "& family")

If our set of Kings is unconnected from all the others then I would assume Peter King and family died soon after arrival (as was the case with most of the whites that went to Jamaica) and that may be the land inherited by the brothers John and Francis King

There's nothing worse that trawling lots of records with many mentions of the King and Kingston for someone called King! ... I think that's one reason why I dived into the French people, three families with that ever so rare combination of the letters "GOU" in their surnames

I've seen some references suggesting that John King may have been born in Jamaica, in Manchioneal - that at least appears to be the primary area that they inherited property

Some records link to King christenings in Stretton, Rutland
Ballantyne lore has it that the Kings were of East Riding - between Scarborough and Whitby

Aha, "Cousin Katie" and James Dunn the architect?

... That's right 

I've not really found anything related to "Kell", useful to know it is a Yorkshire name
I've been using Bing AI for a lot of queries, it's a quicker way of find things but it threw a sulk when I tried to get it to search "Kill King" ... I'm probably on some terrorist watch list now!

I think the original records that I found on Family Search were created by Chris Codrington, as such the records for Dr John Kell King and Francis King already existed. I don't know if John King had Kell as a middle name but using the "Kell" moniker helps differentiate him from the other John King's in the line. I do have a copy of Dr John's will, that was the evidence that confirms that John's brother was called Francis

Kell remains a mystery name, it could be a maiden name somewhere up the line but at present we have no idea who John and Francis' parents were. I have found explanations for all the other middles names, Bryan, Bathurst, Smith, Stapleford and Rodon.

The Bryan, Rodon and Smith surnames are also prominent amongst the earlier Kings that I found

I've not found any direct (marital) links to the Cargill name, but in John's will he does refer to his very good friend Richard Cargill.

I believe that John King's first wife was Louise Henriette Corre Desgouttes, (the surname usually appears as just Desgouttes, but the full name is "Corre Des Gouttes"), daughter of Louis Corre Desgouttes and Victoire Michelle Aimee De Gournay. Victoire was one of the 10 children of Michel Raphael Issac De Gournay and Perrine Chevolleau. Another daughter, Marguerite Antoinette De Gournay married Louis' brother Charles Desire Corre Desgouttes

I'm not sure what happened to Louise Henriette Desgouttes - She was married to John King by 1822  and he married Anne Catherine Norman in 1836. Being catholic, I don't expect they divorced but I haven't found any death record. Her siblings ended up in New Orleans.

Michel and Perrine De Gournay's son, Pierre Bernard De Gournay is registered as the owner of a Stoney Hill Pen, Saint Andrew from 1821 to 1823. He married Maria Josephe Sophie Raccine.
They had two daughters, Annette Elizabeth Bernadine De Gournay and Eleanor Pauline De Gournay.

Annette De Gournay married Samuel Kell King, son of Dr John King
Her sister, Eleanor De Gournay married King Charles Williams, grandson of Dr John King
King Charles Williams was the son of Charles Williams and Mary King
Samuel Kell King was registered owner of a Stoney Hill in 1826

John King isn't registered for Stoney Hill until 1831, possibly when Louise Henriette Desgouttes died? The family history re the John King inheriting Stoney Hill via his first wife isn't the whole picture!

I thought the following was convincing evidence of the connection between John King and Louise Henriette Desgouttes

Marriage of Louis Marchand and Pauline Chevolleau 12th January 1822
Amongst those also signing there are L. H. King, L. Desgouttes, Victoire Desgouttes and J. P. Desgouttes.

The most likely person for L H King would be Louise Henriette Desgouttes, using her married name at the start of 1822

L Desgouttes would be Louise Desgouttes and Victoire (De Gournay) his wife - parents of Louise H
J P Desgouttes would be Louise's sister, Jeanne Pauline b1805

Marriage of Jeanne Desgouttes and Jean Branday - 3rd May 1828
The additional signatures include L. Desgouttes, Louis Desgouttes and John King

They would be Louis Desgouttes, the father, Louis Honore Desgouttes, the brother and John King, the brother-in-law.
Where was Louise Henriette King - it's her sister's wedding
Louis's wife, Victoire De Gournay died in 1826

 

This is the record for the John King that married Anne Catherine Norman

https://www.familysearch.org/tree/person/details/L2HX-JGC

It's FamilySearch.org, which thankfully is free to use

The Blake family tree suggests that it was Anne Blake that married John Norman
Anne was a daughter of Robert Blakes' brother, Humphrey Blake

There is evidence of Robert Blake, his brother Humphrey and John Norman all being involved in Bridgwater religion

Humphrey and Robert Blake and the town's recorder, Sir Thomas Wroth, were members of the (Presbyterian) classis in 1647, and John Norman, vicar 1647-60, was associated with a Presbyterian meeting from 1662

The Norman family tree shows John Norman as having married 1) Alleine and 2) Elizabeth Blake

The Dictionary of National Biography has this to say

He (John Norman) was the bosom friend of Joseph Alleine, the ejected vicar of Taunton, whose sister Elizabeth seems to have been his first wife. Norman was probably the 'Pylades' to whom Alleine, under the signature 'Orestes,' wrote a very remarkable 'Letter from Bath' on 12 Oct. 1668, smoothing over some 'jealous passages' which had occurred between the writer and his old friend and 'covenant Pylades' (Life of Alleine, 1822, p. 432, letter xxxvii.)

Is it possible that the Norman family tree got the details wrong, after all Elizabeth Alleine is only referred to as Alleine and the Blake tree suggests that it was Anne Blake, not Elizabeth Blake

The Dictionary of National Biography also states

He (John Norman) died at Bridgwater, and was buried at St. Mary's on 9 Feb. 1668-9. His wife Elizabeth had died in 1664, and he seems to have married a second wife, who survived him A son, John, born in 1652, matriculated from Exeter College, Oxford (8 May 1669). Henry Norman, master of Longport grammar school from 1706 to 1730, may have been the minister's grandson.