You have an established relationship with the child, and being named on her birth certificate means you have parental responsibility - the same as her mother has. The mother would have to apply to the court to have your PR terminated on the basis of the DNA results if she did not want you to share PR anymore. However, given that she did not do so three years ago when she discovered that you were not the biological father, it is not definite the court would grant this order.
In any case, whether you have PR or whether it is terminated - you have an established 8 year relationship with this child who sees you as her daddy, and therefore the court will do what it can to promote and maintain that in the best interests of the child. Barring any safeguarding issues.
Firstly, the best route would clearly be to try to come to some agreement with the mother, however if she chooses not to agree, or is too restrictive on what she allows, you do have recourse to the courts.
You are obliged by law to try formal mediation to see if that will resolve matters, but if mother won't engage, then you needto attend what is known as a MIAM (mediation information assessment meeting) to get your form signed by a mediator so you can apply to court for an order instead. There are some exemptions from attending a MIAM – the form provides more information on whether you meet that criteria or not.
The form you need is called a C100, which you can find here: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/734888/C100_eng_0818.pdf
The form needs to be a) signed by the mediator, b) filled in by you, (you are asking for a child arrangements order) andthen c) copied three times (with a fourth copy which you will keep for your own records). You must then take orsend the original plus 3 copies to the family court closest to where the child lives.
You can find the right court here: https://www.gov.uk/find-court-tribunal
The fee for filing is £215, but you may be entitled to fee remission if youare on a low income or benefits, see here: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/634329/ex160-eng.pdf
Does that help?