'Ciril' is the later spelling of this river's name, used throughout the text of The Lord of the Rings. In many older editions, however, the large-scale contour map given at the beginning of Book V shows the older form of the name, 'Kiril', which reflects its correct pronunciation.
Notes
1
The highly detailed map of Gondor included in The Lord of the Rings seems to show Calembel on a hill some miles from the fords of Ciril (as shown on the map for this entry) rather than directly on the river. However, Tolkien noted several times (most prominently in his detailed index to The Lord of the Rings) that Calembel was located actually on the fords.
We might reconcile these sources by imagining that Calembel was rather larger than the single point marked on the map, with a major town on the hill, and outlying parts of the township running down from the hill to reach the rivers' fords.
2
Tolkien addresses the meaning of the name Ciril, but is himself uncertain about the name's origins. He suggests a connection to cir-, 'cut', specifically in reference to the river running through a deep rocky channel on its upper course through the White Mountains. This presumably refers to the short northern stretch where the course of the river ran eastward through the foothills of the mountains before turning southward and flowing on across Lamedon. (Somewhat confusingly, Tolkien says in his comments that this channel ran westward, not eastward as shown on the maps - presumably he meant that it ran west to east, or something of the sort.)
3
Specifically, the older spelling Kiril appears on the highly detailed map of Gondor and Mordor in older editions of The Lord of the Rings. This spelling belongs to an earlier phase of composition, and when Tolkien came to prefer the spelling Ciril in the text, that change did not find its way onto the map. The pronunciation is identical regardless of spelling, and later editions of the book have updated the map to use the standard spelling in the text of the book: Ciril.