Shine a flashlight along the underside of the roof and look for a hole in the roof as well as areas that are wet or have been.
How to stop a slate roof leaking.
You might find problems removing the nail with the slate ripper.
To do this use the slate ripper to rip off the nail that hooks the slate onto the slate roof.
Begin soaking low on the roof just above where the leak is visible inside the house.
Have your helper stay inside the house waiting for the drip to appear.
For example soak the downhill side of a chimney first then each side then the top on both sides.
Details are at slate roof leaks temporary repair.
You could make a temporary patch using stick down materials or on some roofs i ve seen roof mastic used in that application.
Insulate the roof if you are going to finish the attic if not insulate the ceiling.
Beware of using asphalt roof mastic on copper flashings it fixes the leak in the short run but corrodes the copper in the long run.
Once you ve found the problem slate you need to remove it.
Start low soaking the area just above where the leak appears in the house.
With good wire cutters you might also be able to cut off the head of the nail which will help.
Run water over the low side of the penetration for several minutes while the helper watches for any water droplets.
Apply more tar over blocking material especially on the edges.
How to repair a leaky roof from inside the house.
Use enough water to mimic heavy rains.
Apply tar to the area and press blocking material into the layer of tar to cover the leak.
Go nuts with the vapor.
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.
All roof leaks must be fixed from the outside.
Insulate the smallest space possible.