It depends entirely on the specific terms of your insurance policy (Nintendo's official warranty extension or a third‑party plan like SquareTrade/Asurion). Nintendo's standard 1‑year warranty does not cover accidental damage.
Without Insurance: Nintendo's official repair for DSi (long discontinued) is likely unavailable. Third‑party repair shops ($40–$80) or DIY shell/screen replacement ($15–$30 parts) are your only options.