State laws differ regarding when or if your driver's license will be reinstated and no longer put on "hold" after you are found responsible in an accident and are being required to pay for the damages you caused due to not having insurance to cover these costs.
In some states there may be a time period which even if you do not take care of repaying the person's insurance company that had to take care of the costs of the damages and then subrogated with you for the monies they spent. In other states your license can be put on hold or suspended/revoked until you pay the person's insurance company back for the monies they paid out for the damages you caused or make a repayment plan that the insurance company agrees and asks the Department of Motor Vehicles (DMV) to lift the hold on your license.
For example in Ohio if you are involved in a crash without insurance, or other FR coverage, you may have a security suspension of your license for 2 years or more and a judgment suspension for an indefinite period, until damages are paid.
In the state of Rhode Island not only your license can be suspended or put on hold. In RI the license, registration, and nonresident's operating privilege shall remain so suspended and shall not be renewed, nor shall the license or registration be thereafter issued in the name of the person, including any person not previously licensed, unless and until the judgment is stayed, satisfied in full, or in approved installments, and until the person gives proof of financial responsibility.
In FL, the Florida Department Highway Safety and Motor Vehicle (HSMV) and the FL Bureau of Financial Responsibility both get involved when an uninsured party is at fault in an accident causing injury, property damage or both. If the person did not have this coverage, they must purchase it and have it certified on a SR-22 form for 3 years. The at-fault party must also obtain releases from the victims that they were compensated fully for their bodily injuries and/or property damages.
The above requirements must be met before the suspension date in the inquiry notice to prevent suspension of the owner's tags and registration and the operator's driver license. If you had no car insurance at all and receive a judgment against you, the FL Bureau of Financial Responsibility will suspend your license, tags and registration for up to 20 years or until the entire judgment is satisfied.
If you cannot pay or agree on a payment plan with the other driver's insurer then most states will suspend your driver's license as a way to try and get you to take responsibility and pay for your actions. Some states may reinstate your license after a certain amount of time has passed while other states will suspend your license indefinitely meaning that until you pay or make approved arraignments to pay they will not reinstate your driving privileges
Most insurance companies will subrogate with an at-fault party and thus want repayment for the monies they spent on a claim that was not their insured's fault. This appears to be your situation and so likely to get your license reinstated you will need to pay in full the insurance company that has requested repayment for the cost of the repairs they paid out for the damages you caused or at least set up a payment plan. If you do set up a payment plan keep in mind that if you do not keep up with the schedule of payments your driving privileges may again be taken away.
You can contact your state's Department of Motor Vehicles to find out more about your licensing situation and find out how long your driver's license will be placed on "hold" if you do not pay the other party's insurance company the monies they have requested. The DMV should be able to tell you what specific steps are required of you to get the hold taken off and your driving privileges reinstated.
As you realize now it is important to have at least your state's minimum liability limits on your vehicle so that your insurance could cover the damages (property damage or bodily injury) you may cause to others in an accident and not leave you personally responsible to pay them using your own assets. Get auto owners insurance quotes here so that when you do get your license reinstated you will be properly covered on the roadways.