How do I get my doctor to admit me to the hospital for back pain?

Don't try. There is no need for a hospital admission to treat back pain. To solve the problem, find an experienced pain specialist and get it treated.
Emergency Room. If you have severe pain that has failed all out patient conservative treatments go to you er and have you doctor admit you for further evaluation. This can be x-rays, MRI studies, certain medications that are given only in a hospital setting. Get pain management to see you, neurology consult etc. Just because you have back pain does not mean you "can't " be admitted. Maybe find a new dr.