Will an insurance company pay more than the policy limits?

My daughter and I were injured in a motorcycle accident a few months ago. (I was NOT at fault). The at fault driver ran a stop sign and was cited by the police. The insurance limits he has is a 50/100/50. My daughters medical bills were about $5000, property damage is about $3000, but my medical bills are about $75000.00... I am seriously injured. Since his limit is $50k per person I'm not sure who is going to pay the difference of the medical bills not to mention pain, suffering, lost wages, etc. My medical insurance is paying the bills but they want to be reimbursed. My auto insurance is not going to pay because I made a stupid decision to drive an uninsured motorcycle. Will the at fault driver's insurance company pay more? I know I could sue, but you can't get blood from a turnip...If I were to sue, what would be something reasonable to ask for and expect to get? We live in Tennessee. Any advice would be appreciated.