uhhh i might be wrong but: female lead wrote down her plans, as well as the plan to break the white haired dude and the villain's engagement, thus when the villain knew what female lead was trying to do, she immediately offered an engagement to the white haired dude. the engagement between white haired dude and the villain is beneficial to white haired dude is because since she is still more powerful than white hair dude, marrying her means she is now his "property" (i think) which inherently increases white haired dude's power in general.
Not his property, but as his wife, he would have a say in what she does with her power, though not entirely. Like how spouses have a say in finances. He would also be able to keep a closer eye on her.
But personally, I think it's all pure trap with no substance. She's poking his buttons to make him feel uneasy and he's not thinking clearly. She's giving out flimsy reasons why marrying her would protect the princess because it doesn't matter what she says so long as he has the idea that marrying her will protect the princess. He's already willing to do whatever it takes to protect her, and that would probably include marrying his enemy.
She randomly wrote the plans didn't she? Most were about her feelings. About the villain... it was just king won't marry her. So how exactly it helped that b****?