Hi Dwayne61
You already got plenty of useful suggestions... So I just wanted to add my thought, even though I don't know if it will be of any help...
I was wondering... Are you sure that you asking if she wanted you to pay for her lunch is the reason of her odd behaviour? It seems very strange to me. You describe your friendship as a very solid one, I find it hard to believe that such a trifle could shake it like this. You even apologized and she said it was okay... I read you're writing her a letter. Well, I wouldn't insist on this if I were you. Telling her how you feel and asking what's wrong is a good idea imo, but I wouldn't mention what happened that day at lunch. Just my 2 cents.
Good luck