Have you asked her directly as to why she is in a bad mood or seems annoyed? Perhaps there is something she wants or needs to talk to you about, but is waiting for you to ask?
Given that you said her "Mum didn't want her", I imagine, would be quite stressful for her and make her feel sad and rejected. Maybe you too are feeling a bit rejected because she doesn't laugh at your jokes, or is unable to appreciate your effort to engage with her.
On another note, maybe you could think about taking up some sort of sport of hobby, and start to engage with other people that you share a common interest in? That way, friendship could well be a natural follow-on, or 'by-product' of the sport or hobby itself. Playing a sport is great for your mental health, as well as your physical health.
And finally, although it's nice to be friendly with people, it doesn't necessarily mean that everybody is our actual friend. For example, I have two older sisters, but I am much better 'friends' with my middle older sister, than my oldest sister. My oldest sister, I feel, has always been a bit cold and distant from me. There's 6 years between us too, so when I was born, she was already in primary school. And when I got to primary school, she of course went on to high school, so yeah, I was always the 'baby' in her eyes.
I do hope that you soon find a friend that you can connect with and have fun with. Like I said, maybe the key is to find a hobby or a sport; something that is a shared interest that you can focus on.
Take care. Still here for you. xo