You can send the gift to her. That way she could still have it. And then you wouldn’t have to go to her home uninvited. From what I understand, you do have contact with her once in a while. That can be enough too for friendship.
It’s better to have friends that give you space to breathe, instead of friends who want to be with you too often. Personally, I like to see friends once or twice in every one or two years. It’s when they break the contact completely without telling me why, when I don’t like it. If people don’t want contact, it’s oké, but they should at least give a reason. But your friend hasn’t broken contact without giving you a reason. You still are friends, as long as you have contact with each other once or twice in every one or two years.
It reminds me of a Pakistani family friend. Nice lady, but too sticky. This you will find creepy and then you will be happy with your friend:
I have sent her salaam through my mother a few times. Then later, I also sent her many times salaam and Eid wishes via her daughters email. For me, that is contact. That is enough. I didn’t visit her often. The last few years I didn’t go to anyone. But she is the only Pakistani nonrelative, whom I have been sending salaam via email regularly. So for me that is contact. Yet, when we met a few times, she kept saying “you have forgotten me, you don’t care about me, because you don’t call me and you don’t visit me” 
These comments irritated me a lot, because if I didn’t care about her and had really forgotten her, then why the hell would I send her salaam every year via email? She also kept saying, you must visit me, you **must **call me, you have to come.
Those kind of sticky friends are just creepy. I hope this will help you to see the difference between your friend and mine. (I still consider her a friend, even though I was so annoyed I told her I wanted no more contact)
Your friend is normal. She isn’t too sticky, she doesn’t want to be with you too often, she gives you time to breathe. I think in her own way she probably does care about you.