How do we know that Cordelia truly loved her father?

Cordelia is the only one of Lear's daughters who refuses to play along with the public charade in which they're all expected to proclaim their love for the old king. Of course, Cordelia's refusal to flatter her father in this way isn't of itself proof that she loves him. But it's telling nonetheless that the only one of Lear's daughters who does truly love him is reluctant to proclaim her love so publicly. As ensuing events demonstrate, Cordelia believes that love is something you show, not something you say. Talk is cheap, as Regan and Goneril's subsequent behavior attests. They have no hesitation in proclaiming their love for Lear, but once they safely get their grasping hands on his kingdom, they proceed to treat him with contempt and disrespect.
Cordelia shows her love for Lear in a number of ways. First and foremost, she marries the King of France and sets about raising an invading army to defeat her wicked sisters and restore Lear to the throne. If we cast our minds back to act 1 scene 1 we'll recall what Cordelia said to her father:

I love your majesty. According to my bond, no more nor less.

Her bond is not only that of daughter, but of royal daughter no less. And it is in this capacity that Cordelia spearheads an invasion force to try and put her father back upon the throne he so foolishly vacated. When Cordelia's plans come to nothing, she chooses to share her father's fate in prison. There, they will sing like birds in a cage. Cordelia has sacrificed everything for her father, including her life, and Lear is most grateful to her for this. If such selflessness isn't the ultimate act of love, then it's difficult to know what is.