It’s always a smart idea to offer to write a letter of recommendation for a colleague who you know is well qualified for the job in question. Consider it good career karma; after all, you never know when you’ll need a letter of recommendation for yourself, and your colleague may be happy to return the favor.
Use a business closing and sign your name. Above all, be professional. If the you’re sending a physical letter, print it out and sign it by hand. Otherwise, just typing your name will do.