You will need to contact the other party's insurance company to see what coverage the at-fault party has to take care of the physical damage to your vehicle. Since you live in the no fault state of Minnesota you are required to have car insurance and included in that is personal injury protection (PIP). This is the coverage, if you had been properly insured, that you would claim your injuries under.

With the no fault laws in MN, injuries are normally claimed under your PIP benefits with your own insurance carrier. So it is likely you will have to take care of your own medical expenses since you were uninsured. Minnesota laws might allow for you to claim personal injury against the other driver but they the medical expenses would have to be high and typically your injuries would have to be serious or permanent. You can seek legal council to help you understand the Minnesota no fault laws more in depth.

As for driving without insurance, if you were cited you will need to take care of that violation. Generally, MN requires you to pay a fine and and proof of insurance at least, the penalties could be even stiffer. Make sure to get Minnesota car insurance as soon as possible.