Good evening Ken,
I'm Doug, and I'm very sorry to hear of your situation. My goal is to provide you with excellent service today.
Yes, you can easily get her into the system. You have to forget about the state child support agency. They cannot force a mother to seek child support.
All you need to do is file a petition to set child support with the family law court in the county and state where your daughter is living. The court will force the mother to accept child support.
In the meantime, DO NOT pay her in cash, but only by check so you have proof of payment. That way you won't be hit with a large support arrears order and the court will see you as a good father, trying to do the right thing.
At the same time you can ask the court (by also filing a petition for child custody/visitation) to order visitation for you so that you can maintain your relationship with your daughter. If you live anywhere near your daughter, you should expect to get visitation of about 90 to 120 days/nights per year.
You would be better off having a local family law attorney do this for you---but if you have more time than money, you can try and do it yourself. Virtually every county has a law library. Ask the law librarian to show you where the books on Pleading and Practice are located. Generally there are many volumes of Pleading and Practice books from which you can pick the proper one for your situation. While the librarian is typically not a lawyer, and cannot practice law, most are quite willing to assist you in locating the chapter in the Pleading and Practice book which you will need.
Once you have found the appropriate pleading, you will note that in addition to the sample pleading which contains all of the necessary structure for your Pleading, there may also be suggestions to make your pleading more persuasive. Prepare your pleading in accordance with the suggested format in the book, inserting the facts of your case as you know them to be. Then all you will need to do is to draft the pleading on your computer, print it out and file the original with the court while serving the other party, or their attorney, with a copy.
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I wish you and yours the best in 2016,