This is the best point of view for a photo without getting the ladders out

Our extension (on the right) was built in the mid 90s, next doors extension went up a year or 2 before.
Both roofs currently drain into a single standard half round gutter which fills up with moss and then overflows down the original back wall of the house and soaks the wall in the bathroom.
The logical solution as far as I can work out, would be to build a proper valley at the bottom of the roof, either from lead or zinc. Ideally I would be able to get up a ladder once a year and sweep it out, so would be good if the bottom was wide enough for a broom, and I could then sweep it from the ladder.
I'm in the very fortunate position of getting on well with my neighbour, and he is more than happy to assist me in fixing this. He provides labour and I pay for the materials (seems fair, it was our extension that has caused the problem)
From the front to the back of the existing gutter is about 13 foot. How much fall am I going to need over the length of it, and how close to the bottom of the roof should I put the valley?
Would you go for lead or zinc? And presumably I would need to make a wooden tray to sit underneath the lead/zinc?
Any thoughts much appreciated