You can either finish it and focus on the reason you delayed and the fact that you are getting inpatient treatment.
Or you can wait and go to the hospital. You probably won’t be in the hospital more than a week or two. The professionals there might be able to help you.
I think the second option is safer.
If you are depressed and having suicidal thinking it is urgent that you get to the hospital. I would wait until you are feeling better to get the appeal done, but mention it to counselors so they can communicate with the school.