I would sign it Sincerely XXXXX XXXXX Wishes you don't want to sound like you still have feelings or want a relationship with them unless you are writing to try to get another chance then you would sign it love or getting closure then you sign it best wishes.
In this instants I would sign it Best Wishes Always and then your name. This way they will know that you wish them well and are thinking about them and care about how they are doing.
I wouldn't make the letter all about love and being with her I would leave it at asking her how she is that way she may reply and then the lines of communication are open. Add the card and leave it at that until she contacts you, then ask to have lunch or dinner and then tell her how you feel about her but you may have to start all over as friends in order to regain her trust and attention.
Yes you don't want to come on strong just yet try to gain a friendship and then see where things can go, if you don't handle this right you may chase her away and you don't want that and you definitely want to take thngs slow and talk about getting another chance when the opportunity presents itself.