Details are at slate roof leaks temporary repair once the necessary slates are removed the valley is usually replaced with new heavy copper or on some roofs folks use other flashing materials like aluminum i don t recommend that.
How to repair a slate roof leak.
With good wire cutters you might also be able to cut off the head of the nail which will help.
You can use either a dry bonding gutter or the standard bonding gutter.
I ve repaired complex metal roof flashings with torch down modified bitumen but there is a fire risk doing so.
To do this use the slate ripper to rip off the nail that hooks the slate onto the slate roof.
You might find problems removing the nail with the slate ripper.
Or see these.