Hello, my name is Marcus.
You can make an application under section 56 family law act 1996.
The application is for a declaration of parentage. Applications are often made when a father is refusing a DNA test and the court consider all evidence available such as evidence from the mother that they were in a relationship of the time and that he is believed to be the father. The court if satisfied can then make a declaration that they consider the alleged father to be the father.
In your case this is not entirely helpful although the court could make a declaration that they consider you are the father it would simply be based on all the circumstances of the case and in particular your evidence but at the time you were in a relationship with a mother and therefore most likely to be the father.
If an application is made however then the court can encourage your daughter to have a DNA test as it would resolve the issue once and for all.
However if your daughter refuses then the court can draw an inference from her failure to cooperate and make a decoration for your her father. As I say that that is not entirely helpful to you as you want to know whether you are actually the father or not rather than be successful in securing a declaration that you are the father due to your daughter's refusal to undergo a DNA test.
Do you think that if it is determined that you are the father then your daughter would wish to have a relationship with you or do you think too much damage may have already been caught by her mother.
There is one of the avenue could possibly explore and it is to obtain a copy of her birth certificate. This should be possible as it is a public document.
If your name has been removed from the birth certificate and another person's name placed on the certificate as father then in all likelihood a DNA test has been undertaken revealing that the named person is in fact the father.
A birth certificate can only be changed with evidence by way of a DNA test.
If the birth certificate remains with your name on his father then I would suspect that no such DNA test exists as one would have thought upon such a DNA test result revealing the other person is the father the first thing they would have done would have been to change the birth certificate.
I do hope that this helps a little.
If you have any further questions please do not hesitate to ask.