The answer is here:
Dr. David Kelly
Mr. David Jones: To ask the Secretary of State for the Home Department (1) on what date the police inquiry into the death of Dr. David Kelly was closed; 
(2) on what date the police investigation into the death of Dr. David Kelly was closed. 
Hazel Blears: Thames Valley police were investigating the death of Dr. David Kelly on behalf of the Coroner for Oxfordshire. The Coroner declared on 16 March 2004 that he accepted Lord Hutton's conclusion and that he would not re-open the Inquest.
The answer from Hazel Blears is an interesting formulation.
It avoids actually stating whether the Thames Valley Police had closed the investigation into the death of David Kelly and, if so, when.