It was somewhere outside Barstow when (S
Burkey) wrote:

Had the chimney sweep come last week and now we're trying to figure
out how to clean the brickwork at the back of the inglenook.

So how would be the best way to tackle this?

I'd expect to need to re-point after cleaning. A lot depends on what
type and condition of brick you're dealing with. You really need to
try sample areas with your cleaning technique before attacking the
whole lot.