how to fix valley guttering correctly?

Can't say I've ever come across a valley between two houses. The only set up i've seen is a box gutter at the eves between two houses.
Without seeing it I can only guess.
You can't unless you know what you are doing because to replace a perished old valey means stripping off the tiles either side. Removing the valley and fitting a new one, then replacing/re-newing the tiles. It's a major job.

As it appears from your description to serve both properties then it will be a split cost so have words with the neighbour.

However you can coat the exsiting with Aquaseal No5 bitumen paint about 3 coats and repoint the cement under the tiles [if any, some are open edges, but this lasts about 3yrs before a repeat is needed.

Bob
September 2009