With good wire cutters you might also be able to cut off the head of the nail which will help.
How to repair a slate roof leak.
You might find problems removing the nail with the slate ripper.
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.
Installing a bonding gutter this video shows a roof repair and the fitting of a bonding gutter between a slate and a tiled roof but it will also fit two slate roofs together as well.
I ve repaired complex metal roof flashings with torch down modified bitumen but there is a fire risk doing so.
Or see these.
You can use either a dry bonding gutter or the standard bonding gutter.