What do I do if I’m being sued for a personal injury regarding a car accident from over a year ago?

Your question includes many sub-parts.  First, a public defender applies to a criminal case.  You have described a civil case.  That is why you cannot get a public defender.

Second, did you have automobile insurance at the time of this crash?  If so, problem solved.  Call the insurance company, tell them what is happening, and they will defend you.

If you did not have auto insurance, then you have some problems.  I do not represent you and I don't even know what state you live in, so I cannot give you specific legal advice.  However, I can tell you some general things:

It does not matter that the crash occurred a year ago.  The statute of limitations is probably much longer than one year.  In Florida, people have 4 years to file suit.

I do not know what you mean "they are suing for $200,000"?  Do you mean they filed suit in court for this amount?  Or, do you mean they wrote you a letter and demanded $200,000?  Either way, you are going to need help to sort this out.  You are going to need a lawyer.  Nonetheless, just because they say they want $200,000 does not mean they are entitled to that amount or even that they can get it.

Your question does not contain enough information for me to know what the girl and her mother might be trying to do.  If you were at fault for the crash and do not have insurance, it is possible that they are trying to collect from their own insurance (under the uninsured motorist's coverage).  If so, they HAVE to go after you.  They have to show that you are liable and you have no insurance or assets before they can recover from their own insurance company.  As you can see, this can become very complicated.

If you have no auto insurance for this crash and you have assets, you definitely need a lawyer.  I strongly encourage you to pay a lawyer in your state to listen to all the facts and advise you about what you can do. 

I STRONGLY ENCOURAGE YOU TO RESPOND TO ANY LAWSUIT filed against you.  If you do not respond within the time listed on the Summons, the court can enter a default judgment against you.  This means you lose.  And, you will have to pay damages.  If you do not respond to the lawsuit, you will not even be able to show that the girl's claims are bogus or that they worth far less than $200,000.  If you have been sued, it is a very bad idea to ignore this.  Even if all you do is write a letter to the court to say that you are not responsible for this and you have no lawyer, that is better than doing nothing.

I wish you luck.

First of all, the Public Defender's office only provides legal representation for indigent Defendants in criminal cases.  You indicated that you are being sued as a result of an automobile accident.  This is a civil case.  Second, if you had automobile insurance coverage at the time of the accident, you should notify your insurance company immediately.  As part of your insurance policy, your insurer should provide legal representation for you in the event you are sued for an automobile accident.  If you did not have insurance at the time of the accident, then you should retain the services of an attorney to represent you.  Additionally, you did not indicate whether or not you were cited by the investigating police office (if reported to the police) at the time of the accident.  If you received a citation as a result of the accident, then you may be entitled to representation by the Public Defender's office on the traffic citation only.