Biscuit... mate... I'm so pleased for you and this new lass in your life and her daughter.
I will now be bold: and I'm normally bold anyway !
Spend time on your relationship with the new lady in your life and let the realtionship with her daughter occur naturally now.
The reason I say this is ... organic and natural realtionships work from happy parents most times,
I met my wife, and fell in love with HER without needing to impress her with my parental skills.
THEN we had a kid. We became parents after falling in love.
I know for you it's about making sure she knows you're ok with her being a mum, and franklly... I think you have.... but now, its you and the new lady now bud.
I don't mean ignore the little girl.. you know that. And I think you'll find kindness natural
tbh....be nice to the little girl, and be the best you can be.... always (I kinda know you will actually- you're a nice guy)
But now make
your relationship with your new potential woman work. The kid will, literally, be happy for you being her mum's best mate.
One day, if you all gel, you'll end up telling that little girl off and then reading her a story.. and taking her to school maybe, sittin her on the naughty step, then planting seedsin the garden within an hour. Kids moods flex like a fishing rod.. they really do. But you need now to concentrate on your bit of it