I find it interesting that so many here assume that she's rejecting a good dad in Boel for this pipe dream in Albert. Boel certainly bankrolled her lifestyle for a long time, but outside of financial matters, I would be shocked if they had a good relationship, given the timeline of established facts in her life.
No matter which man was her father, she was born during a time when her mother was having a long-term affair. Her mother's marriage to Boel officially ended when Delphine was 10, but several years prior to that they'd already moved to England to live with the man who would become her mother's second husband. At no point in that scenario does it seem likely that her mother and Boel would have had a good relationship. If anything, it was likely either (a) tempestuous or (b) cold and distant. Given that Delphine was living primarily with her mother, it's just not a recipe for a her establishing a solid footing with her legal father, no matter what checks he was willing to write.
And if, all the while, her mother was constantly whispering in her ear that her real father is a prince...
That turbulent childhood plus her actions now paint a picture for me of a woman who felt emotionally injured by/distant from her parents (or at least her legal father) and who is holding on to the dream that establishing the biological facts of her parentage will help her put that hurt to bed.
Now, I don't think having Albert named her father will actually be of any help to her. I think she's holding on to some impossible dreams and unrealistic expectations. The heart doesn't magically develop the ability to heal because some court changed a legal status. Boel may have loved her more than she was willing to acknowledge, but she's destroyed whatever was there; meanwhile, nothing can make Albert suddenly become a warm presence in her life. She'll walk away from all of this the same wounded animal of a woman she was to begin with, only with more notoriety and less money.
It's a sad situation all around. I feel bad for everyone involved.