Slate is a fine roof that can last a lifetime, but needs occasional maintenance or repair. Your solution would certainly leak and would likely damage the roof causing you much greater expense. Hire a contractor with experience repairing slate roofs. Slate is not generally a DIY repair due to roof pitch, fragile material and specialized expertise. A contractor will have the right ladder and can obtain a piece of replacement slate. Have the rest of the roof inspected while he is there.