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