If you decide to give this a time limit, then you can text her until that time limit is up. It would be best to text her often and avoid having her do all the texting.
If you want this girl to go out with you on a one on one basis, building a stronger rapport with her over text and even phone calls would work to your benefit.
Keeping this in mind, if she were to text you, make it a meaningful conversation that is full of substance and laughs instead of keeping it quick.
If you can get her to trust you over time by building this kind of rapport with her, she may eventually be willing to venture out with you on a one on one basis.