YA girl runs away to private island but it's abandoned. Losses bunch of weight. survives winter

From 80s maybe early 90s. A young girl from a wealthy family runs away to (I believe) I family owned private northern island that supposed to have an empty vacation home. The girl is overweight and depressed.

When she gets there, she discovered that the home had burnt down. She decides to stay and tough it out. I don't remember, but I think she was stuck there.

She ends up staying until winter and the water freezes and she is able to walk across the frozen lake to the main land. By this time she has lost so much weight she's unrecognizable.

TIA!