Why is it important to Elizabeth that her father respects Darcy (chapter 59)?
Your question identifies the close bond that Lizzie has with her father. Thruoghout the novel Lizzie and her father are shown to have a closer relationship than Mr. Bennet has with any of his other daughters. In the very first chapter, this is clearly established when Mr. Bennet praises Lizzie´s "quickness of mind" and jokingly comments that he "must throw in a good word for my Lizzie". Lizzie, out of all the children, possesses the wit and intelligence that marks Mr. Bennet´s character, and Lizzie is really the only character that truly understands and sympathises with Mr. Bennet. This is why his blessing is important to her, even if she does not fear his disapproval:
She did not fear her father´s opposition, but he was going to be made unhappy, and that it should be through her means, tht she, his favourite child, should be distressing him by her choice, should be filling him with fears and regrets in disposing of her, was a wretched reflection, and she sat in misery till Mr. Darcy appeared again...
It is clear from what Mr. Bennet says that he above all else does not want his favourite daughter trapped in a marriage (like him) where she is unable to respect her partner in life, and thus Lizzie is able to set his mind at rest, by revealing to him Mr. Darcy´s involvement in the Lydia episode. Having gained his approval, she feels "relieved from a very heavy weight" as now her father, who she loves and respects greatly, now has seen beneath the surface of Mr. Darcy´s proud exterior and respects him.