It is normal for a law firm to be called a LLP (Limited Liability Partnership) For tax purposes the organization is treated as partnership rather than a corporation. The important thing is that ALL of the partners must be lawyers. This business form is used for other professions (Doctors, Accountants etc.)
You will see some professionals use LLC(limited Liability Corporation) but it is not the usual form. Basically it is all tax driven.
Never heard of insurance that doesn't cover personal injury! If that is legal where you live, you really need to move someplace civilized.
Were the police called? Did they make a report? Did anyone receive a ticket?
You need a new insurance company too, your agent is an ***.
Anyway, if the other guy's insurance won't cover it, and you don't want to file with yours, you have to sue the other driver for the damages. That means actual money you are out for medical expenses, lost wages, that sort of thing.
So, if you have lost days of work over this, they count. The ER bill counts. Dr visits count. Once you have it all together, add it up. Compare that number to the max amount allowed in small claims court in your jurisdiction. If you number is less than the max, you sue in small claims. If your number is higher, you sue in circuit/superior/whatever they call it where you are, court.