Dear Eugene, Eugenia?

Sorry I have not written to you. I was a very busy.
How are you? What are you doing? How are you doing in the university? I am sure you learn to perfect! Я полагал бы что-нибудь похоже на I am sure you are making great progress!
I am fine. I am studying English again.

For my Birthday Andrew gave me a marriage proposal! It was the best gift (for me)! I am so happy! You will come to us on a summer wedding? - Во вопросах на английском порядок слов обычно следует как "Will you come to us...?"

for our summer wedding?

we a waiting you! - We await you / We are waiting to see you!
I am sorry I need to put to bed my daughter. - Тут простое дело подрядка слов I am sorry, I need to put my daughter to bed.
You greetings from my mother. - Greetings to you from my mother 

I will wait you answer. - I await your answer (Хотя, по опыту, на английском вообще, с такими фразами в письмах сталкиваемся довольно редко.)
I miss you!
Sveta.