I was unable to find a suitable lance to get to 6 metres (let alone 12 metres) so in the end I recently adapted a water fed pole to apply a detergent via a hand pump up garden spray which works really well using the existing brush head and jets.
I simple cut the trigger mechanism off the lance of a 5lt hozlock garden sprayer and cable tied it to the side of the butt on a water fed pole and then re-attached the pump up tank, its brill! The only additional cost was a micro bore socket to fit the hose tail from Gardiner about £9 so I could quickly separate the tank from the pole and hose.
I believe this would work much higher as there is loads of “oomph” at the brush head, in fact a bit too much I only have to tap the trigger a couple of times before it saturates the wall.
If you think this would be of any use to you, just let me know and I will get some pics of the set up to you.
DP