I think it is really nice you are doing that. I don't think most sponsors would take their role to heart quite that way. I think teaching her songs and traditions are huge. History can be learned from all of the written materials out there.
I think it would be really, really nice if she could participate with a local collegiate chapter to attend a philanthropy event with them and to attend a recruitment workshop just to learn how collegiate chapters do recruitment. It would be great to connect her with a local collegiate chapter either way so she can participate in some short or long term alumnae support, sit in on workshops, assist at recruitment and get a feel for what collegiate life is all about, as well as being active as an alum association member.
Maybe you two can plan an alum-collegiate mixer together.
And make her some silly sorority stuff-- decorate a candy tin or paint a pin box. She may be too old for such things, but it is part of being a sorority girl!