If you want this to work then tell her that you're not up for an on/off relationship.
If she can't give you this, then you have to be prepared to walk away and stop being 'close' friends. By all means keep things friendly, and have fun together if you see each other as part of a bigger group. But going round to stay over is just giving her what she wants without her having to give you any commitment back.
She may just need some time to sort her head out. If you give her some space (proper space!! - go out with other friends, have a good time for a month or so) then she may well come to realise that she's made a mistake by splitting up with you. Space, also means time - if she says she's realised how much you mean to her after just a few days, or a week - it might be what you want to hear, but it's unlikely that anything will actually change.
If you keep trying to be as close to her as you can, then more than likely you'll just build things up for more heartbreak further down the line.
Once she's had time to sort her head out, you have to make sure that she's not just telling you what you want to hear. If she wants to get back together, remind her that you were hurt by her changing her mind before and tell her that you don't want to go through that again. If she can convince you that she is more serious this time around, then tell her that you need to think about it for a while. Let her spend some time showing you that she's serious about you - if she is, then she'll be happy to do the 'chasing' for a while.
More importantly! - If, after some time apart, she says she does just want to be friends - then you have to decide if you can genuinely 'just be friends'. If you cant, and you know you want more, then give yourself as much distance as you can. Do not under any circumstance end up going back to being 'close' friends in the hope that she'll change her mind eventually. You'll never get what you want from things that way.