This is Carpe Diem.
My god, is it easy to live! While I was doing a night shift at a PC room, I decided to play a daughter raising game out of pure boredom and nostalgia. But the ending she wound up with was... an Old merchant's concubine. When she got too annoyed and tried to find a different game to play next, a pop-up screen came up on her computer. [You are invited to our world! A place where you can get anything you want!] As soon as she clicked the yes button, she heard a butler's voice:"It's time to wake up, My Lady." Before she knew it, she had become the "daughter" of the game she just played! Ah, she probably would've chosen a better name if she'd known this. Rollypolly Perero. Why did a pack of cookies have to be in front of her when she typed in the name? Could a 24 year-old trapped in a 10 year-old's body successfully manage to become an adult?
Most of the details from the question match, but there are differences. This is not a reincarnation story and the protagonist doesn't die (at least not onscreen), she's transferred into a game she was playing. Immediately after appearing in the game world, her stress levels skyrocket and she gets sick from that. The original body doesn't die. The MC retains all of her memories from before waking up in the game.
She and her father had a poor relationship. Before the story starts, she's essentially a living puppet with no motivation or personality, which makes sense in the context of a daughter raising game. After the MC takes over, due to her confusion, frustration, and just thinking of everyone else as NPCs, she refuses to call her father "father".
Her father is described as one of the top-four mages. After an incident, he gives her a set of magical earrings that changes her hair color from the initial silver to a dark blue, matching her father. As this is in a game, she still has status and schedule popup windows that no one else can see.