The length (height) of chimney required in any application is typically determined by application of the "3-2-10 rule". In no case may a chimney extend fewer than 3 feet above the point at which it penetrates the roof. It must then extend further, if necessary, to a point no fewer than 2 feet above any part of the structure within 10 feet horizontally from the chimney. So, it depends on your roof pitch and on the point at which your chimney exits the roof. Rick
EDIT: Your existing chimney appears to me, from the pics provided, as though it might actually be taller than it needs to be. But, it's hard to tell from the ground-view angle of the pics.
EDIT: Your existing chimney appears to me, from the pics provided, as though it might actually be taller than it needs to be. But, it's hard to tell from the ground-view angle of the pics.