How the TV station might respond. If we were the lawyers for the TV station and we received your letter, we'd start by talking to the producer of the segment. We'd find out why someone else was listed as the songwriter and not you. It's possible the producer may have information that contradicts your claims. If that's the case and an investigation supports the producer then your letter would not provide satisfaction for you. If the producer can't provide a reasonable response, the lawyer may contact you to get more information (and perhaps settle the dispute). Alternatively, the lawyer may blow you off and not do anything and wait to see whether you will file a lawsuit. (Sad to say, the response might be different if the letter were from a lawyer.) BTW, you should also file a copyright application for the song if you have not done so already. If you plan on soon filing a lawsuit, you may have to take advantage of the expensive expedited application process.
How you can improve your letter. The most effective cease and desist letters are the ones that anticipate all arguments from the other side. So if you have facts that demonstrate that the other alleged composer is not the true composer, you should succinctly introduce your facts and anticipate the station's arguments. You don't need to show all your cards. You can say, "I have facts that contradict any claims that someone else has written this song." Also we know that lawyers spout stuff like "pursuant to Title 17 of the U.S. Code" but we think that's kind of bunk. Why not just say "Under U.S. copyright law ..." Finally, avoid threatening a lawsuit. That may give rise to what's known as a declaratory judgment action and you may not want to be drawn into a court battle. It's best to wind up with something such as, "If we cannot reach a prompt resolution of this matter, I will have no choice but to pursue other legal options."