Any ideas on different ways to inspire a niece and nephew who had a rough week because their mom just couldn't get her act together?

My Brother in law has a daughter whose life he is not involved in. My husband and I would like to take an active role in her life so she can know our side of the family, but how do we go about establishing a relationship with her? She is 7 years old. She and her mother live with her grandmother, who is in control of everything in her life. My husband and I are told we can see her and do things with her, but everytime we try, it never works for them. We are starting to get very frustrated with the whole situation. As my brother in law is an alcoholic, he is only allowed supervised visitation, which he has not seen her in 3 years. We have assured our neices mother and grandmother that her father would not be involved if we are allowed to see her. We just want to be a part of her life and let her know who much we love her.