I don't know what her mother is like, but given her father's problems, her life is likely filled with great uncertainty and moments of great fear. It is hard to let go and just be a kid with all that going on, even when you are at Grandma's and everything is taken care of.
I remember when my niece was this age. If asked, she would not agree to go and do anything. It all sounded like fun until time to do it. Then she backed out. She wanted to do it all so badly, but she was afraid to go and have fun. It was being disloyal to her mother, who was miserable with her addiction and did NOT want her daughter to enjoy being with our side of the family.
My mother finally just started saying that we were doing X on this day and all the kids are going. Niece would often end up joining in and having a great time because she would feel safe enough with us to do so.
I think you can do this with your granddaughter. Don't pack every single day, but pick a few of the fun things and just say that today we are doing X. Make sure it is something that YOU will also enjoy, and go and have fun. Truly play with your granddaughter, and enjoy the time with her. Make those memories and I bet she will have fun with you.
Be aware that your granddaughter may at some point break down and tell you she feels bad for having fun because her parent cannot. That is when you tell them that you are sure that Dad wants them to enjoy this for him, and to enjoy every day to the fullest and to play and learn all she can. Let her know it is hard for you too, and that you miss him. Then after a few minutes, go back to an activity when it seems appropriate so that you don't spend a long time dwelling. My niece used to do this almost every time we took her to do anything remotely fun. We got pretty good at giving her a moment or two and then redirecting her back to the fun.