Missed my flight because I got sick, will the airline refund me?

Sorry to hear that. What I’d try:

1. Contact the airline as early as possible, ideally before departure.
2. Get a doctor’s note or medical proof.
3. Ask for a travel credit if a cash refund isn’t possible.
4. Ask for taxes and fees back on non-refundable tickets.
5. Check travel insurance and card benefits. Illness is commonly a covered reason.

Keep all documents and ask for everything in writing. More about missed flight refund options is explained here.