Most UK training contracts Orrick. Fried Frank. King & Spalding. Payne Hicks Beach. Wiggin. Haynes and Boone. Katten. Winston & Strawn.
The covering letter should be brief and certainly contained on a single sheet of paper. It should identify who you are and record that you want to apply for the position. It should have a strong final paragraph, summing up what you can do for the company.