House of Detention
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The House of Detention was built on the site of two earlier prisons, the Clerkenwell Bridewell for convicted prisoners and the New Prison for those awaiting trial. The Bridewell closed in 1794 and its functions were taken over by the Coldbath Fields Prison at Mount Pleasant. The New Prison was rebuilt in 1818 and in 1847, at which time its name changed to the House of Detention. (Wikipedia)

Clerkenwell 101
38 Clerkenwell was home to a number of prisons. The notorious Fleet Prison was built by the side of the River Fleet in 1197. Other prisons include Clerkenwell Bridewell (1615), Coldbath Fields Prison (1794) and the House of Detention (rebuilt 1845). In 1847 a new prison, The Middlesex House of Detention, was built on the site of two earlier prisons, Bridewell, and the confusingly named ‘New Prison’. It closed in 1890 and was replaced with Hugh Myddleton School in 1892. It is now an apartment building called Kingsway Place, but the original dungeons still remain below. The dungeons and cells provided a safe haven for the schoolchildren and staff during bombing raids in World War 2.
Wikipedia
The Londonist
London Historians

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