The unusual point here is the Fleet was a debtors' prison. It was the venue for marriage of all types of people, both the wealthy and the poor, from soldier to to gentleman, all used the Fleet as a convenient means of marriage without the binds of the church ritual.
It has to be said, therefore, that the Fleet was not a chapel for the lower elements of life, the criminals, but for all sort of people wanting to perhaps have the privacy of the marriage for one reason or another kept secret or just for the reason of pure convenience.
The Fleet began ceremonies as far back as the early 1600s; as the prison became over crowded, the surrounding area of the Fleet outside the prison also housed prisoners such as clergy---or suspect clergy of rather ill repute--who were just plainly in debt, and until the debt was paid they were members of Fleets community. . These clergy performed marriages on couples from London and many from other counties, so it wasn't just Londoners that chose to marry there. Chapels were to be found in the oddest places--well, odd by today's standards, such as Inns outside the prison walls.
When you consider that in the mid 1700's 15 percent of the average 50,000 marriages that occurred in England per year were in Fleet or one of the other chapels that sprung up to accommodate this need, it's clear quite a lot of marriages not going down the traditional route.
To understand the scale of this, from early 1700 to the mid 1700s there were as many as 100 clergy and clerks in and around The Fleet making money from irregular marriages. These marriages weren't seen as shady, disreputable or dishonourable; it was seen as a valid marriage under English common law but was not in accordance with the ecclesiastical law of the Anglican Church.
When Hardwicks Marriage Act took effect in 1753, from 1754 onwards marriages had to be in compliance of the cannons of the church so that they could then be legal under English law. This, then, put a stop to the irregular marriage ceremonies.
The registers of the Fleet survive in part. Some are indexed on the online IGI at www.familysearch.org.
Some have been indexed, while others have no indexes at all. Most of the Fleet registers seem not to be transcribed and are often overlooked when searching for lost ancestors. The trouble being they are also scattered around and not in one archive.
Many Fleet registers have been lost, which isn't encouraging for the researcher. About 290 registers, however, do survive; plus there are over 500 notebooks surviving. The Society of Genealogists in London has an index of about 6,700 Fleet marriages (www.sog.org.uk)
The National Archive in Kew has record in the RG7 series (http://www.nationalarchives.gov.uk/).
The Society of Genealogists also has a booklet;
Irregular Marriages in London Before 1754 by T. Benton.