If your mom come here, you should write and sign the letter by youself.
You just say you invite your mom to come here to see how American lives and see mountains and rivers of US. Do not say you miss your mon and your mon misses you. If you miss your mom and your mom misses you, you may go back to China to see her. It is not reasonable that you let your mom come here to see you in both Chinese culture and American culture.