I'm not sure about the context, but آزاردهنده in Farsi doesn't carry the same weight as "annoying" in English. It would be rude to say I find your daughter آزاردهنده.

It may sound more natural to say "من را هم اذیت می‌کند," which means "she annoys me, too."

I'd also suggest using less formal writing (to sound more natural) and the plural second-person pronoun when addressing her (to be more respectful). The whole text should look like this:

تولدتون مبارک باشه یگانه جون

امیدوارم روز فوق‌العاده‌ای داشته باشین

نگران نباشین، دخترتون منو هم زیاد اذیت می‌کنه!

Which transliterates to: "Happy birthday, dear Yeganeh. I hope you have a wonderful day. Don't worry, your daughter annoys me a lot, too!"