Nice one! I would start by radically simplifying the setting via assumptions:
- Assume sewerage hatchays are only located on streets
- Assume a rectangular structure of the London street network with horizontal and vertical streets intersecting in a 90 degrees angle
- Assume average distance of parallel streets of 200 metres
- Assume London covers a geographical space of 12x12km
- Assume there is a sewerage hatchay every 250 metres on average (4 per km)
The general calulcation then looks something like:
[Total length of London street network in km] x [Number of sewerage hatchays per km] = Total number of sewerage hatchays in London
Hence, you need to to calculate the total length of the street network first:
[Number of streets x average length] = Total length of street network
From our assumptions we infer that the number of horizontal streets is 12x5=60 (since there are 5 parallel streets per km). The same is true for vertical streets: 12x5=60. So the total length of the street network is (60+60)*12km = 1440 km.
Now inserting into our above calculation yields:
1440 km x 4 = 5760
So, given the above assumptions, there are roughly 5750 sewerage hatchays in London.
It is important to also stress that this result is equally sensitive to all assumptions (since all elements are connected via multiplications). Hence, changing an assumption by a certain percentage will directly translate into a corresponding percentage-change in the result.