Lead work around my conservatory - Roofing job in Ilford, Essex

Job description

I have had a conservatory poorly fitting that joins on to the back of my house,and following a dispute with the company that fitted it i don't want them back. Where the roof joins the wall of my house it leaks in heavy rain and water runs down the wall. Im not sure what would be best to do whether to repair existing lead or if it is the lead at all?