Not quite what you are after, but you may find of use in the National Archives WO28/341 “Journal from the 1st of March 1812 to the 29th January 1813” which is the day to day account of the garrison. Unfortunately there is gap from the previous volume, which ends in autumn 1810, and this one.

I would imagine that Cooke's official dispatches will be at Kew as well, but you'd need to have a look through the catalogue - a search for Cadiz within the WO1 series ought to turn something up.

Hope that's some help at least.