I too would discuss a trial period.
Firstly a shortish time, 2-3 weeks (include a weekend at least)
Then maybe a longer period (half a term?) and build in time off for you.
Obviously you'll need some basic rules, and make the point that you may need to discuss some things with her parents before you can answer. I would also be very clear about her chores and responsibilities in your home. It may help to talk some over, then revisit after the first trial period.
I imagine she will still be going home a lot, and will be the responsibility of her parents.
As long as it is kept informal, parents retain responsibility, any financial arrangement is purely expenses, then you can keep it informal. It would be a good idea to let school know 'she's staying with me for awhile', emphasising its temporary nature.
I think we'd all like to know how you get on - I too have known it work extremely well. Grand-daughter gets space, the benefit of your wisdom, learns some skills, and you get a young companion!