Dear John,
Thanks for your letter. I would be glad to tell you something about myself.
I have many free time activities. For example, I play table tennis with my friends three times a week. I also read books every day.
I like watching TV, too. I like soap operas best. It's exciting to follow the stories and see what happens next.
Can you tell me something about yourself? I'm looking forward to hearing from you.
Yours,
Li Ping
Be careful with "I feel excited" . Its meaning is "having, showing, or characterized by a heightened state of energy, enthusiasm, eagerness, etc. : feeling or showing excitement.
For example: The night before Christmas, the kids were so excited, they couldn't fall asleep.
But it can also have a s*xual connotation. So for TV or sports, you would say:
It's exciting to watch the World Cup and cheer for your team.
It was exciting to see the show because we didn't know what would happen next!
And yes, you can say "glad" as you did in your original post. In American English, we would use "happy" I'm happy to do that/I'm happy to tell you, etc. But "glad" is not incorrect.
We usually use "glad" when someone asks for something, and you are happy to do it. So if this letter is in response to someone asking about you, then 'glad' is a perfect word to use here.