Your former partner does not currently have 'parental responsibility' for your daughter (unless you were married to him when she was born).
Quote:
"A father can apply to the court to gain parental responsibility. In considering an application from a father, the court will take the following into account:
* the degree of commitment shown by the father to his child
* the degree of attachment between father and child
* the father's reasons for applying for the order
The court will then decide to accept or reject the application based on what it believes is in the child's best interest".
Source:
http://www.direct.gov.uk/en/Parents/ParentsRig hts/DG_4002954
Based upon what you've written, I can see no way that a court would give your daughter's father parental responsibility.
If you were to die, 'the authorities' (i.e. social services) would be under a duty to find the best way to look after your daughter's upbringing. They would seek to ensure, if possible, that contact was maintained with family members that she already knows. So they'd enquire to see whether your parents or siblings might be able to look after your daughter (and whether they'd make appropriate foster parents or adoptive parents). If that wasn't possible they'd try to arrange foster care for your daughter (perhaps with a view to possible adoption by the foster parents in the future) but while still maintaining contact with those family members that she already knows.
Chris