Hello, I'm a small animal veterinarian and I can try to help answer your question about your kitty.
Sorry to hear she is doing this ... definitely can be frustrating.
At 14 years old, she should probably get examined by her regular vet +/- blood work just to make sure liver, kidneys, blood sugar and thyroid are all well.
However, the half in - half out of the litter box isn't indicative of anything in particular health wise. However, if she is too weak to get all the way into the box for some health reason... the blood work will give us a clue.
Sometimes at 14 years old, our dogs/cats experience some of the symptoms of becoming senile. They forget that they are supposed to get all the way into the litter box for instance.
It seems like she is trying tho
You could try getting a larger litter box. A trick that worked nicely for my cats was using one of the large rubbermaid tubs with a lid. Cut a hole in one side for a door. These come in many sizes ... most larger than even the largest standard litter box.
You can also try putting some dried cat nip into your litter to make it more attractive.
Hopefully some of my suggestions help. A large rubber mat around the litter box that can be easily cleaned if she has accidents also decreases frustration.
Please feel free to respond.