The ancient Fleet river runs through this area of London. The Fleet no
longer runs on the surface of the city, alas - it is one of London's
many subterranean rivers, converted into a sewer in the Victorian era.
It has been completely underground since the expansion of Hampstead -
where the river rises - in the 1870s.
The river flows from Hampstead to the Thames, crossing Camden, King's
Cross and Clerkenwell en route. The elgantly curving Farringdon Road
(which intersects with Calthorpe Street) follows the line of the
river. The river gives its name to Fleet Square (around the corner
from Calthorpe Street), the (long demolished)Fleet Prison, the more
famous Fleet Street, and Fleet Lane - these last two being further
south, by the mouth of the old Fleet.
You can see more details of the course and gradual burial of the Fleet
under the burgeoning metropolis at this site. It has some very fine
illustrations of boats sailing from the Fleet to the Thames.
London's Subterranean River Fleet
You can also read about the Fleet and the wells which fed it at this
site. The Clerk Well, or Clerkenwell, is not far from Calthorpe
The River of Wells
There is a stylised map of underground London here. As the site says,
the river is unimposing now, but "such names as Fleet and Effra still
strike fear into the hearts of underground engineers, who know that
any careful excavations could be ruined by the collapse of a rogue
Heritage: Exploring Underground London
You can see the Fleet on this 1827 map. Calthorpe Street is now the
junction off Gray's Inn Road to the right, immediately south of Acton
Street. You can see the course of the Fleet running off parallel to
the right of Gray's Inn Road, following the line of present day
Farringdon Road. King's Cross was known at the time as Battle Bridge,
after the stand-off between Boudicea and the Romans.
Greenwood's 1827 Map: King's Cross
A recent map is here. You'll notice that the river Fleet has vanished
under several miles of Victorian engineering:
Multimap: Calthorpe Street
In this aerial photograph you can see Calthorpe Street running above
the post office depot at the centre of the picture:
Multimap: Aerial Photo
In short, the Fleet River does run through the area you're interested
in, but it is utterly invisible unless you choose to go looking for
it. Thousands of people who live and work in King's Cross and
Clerkenwell have no idea that the river exists.
I trust this is what you were looking for.
Search strategy: rivers + clerkenwell + king's cross
fleet + london