I am sorry to learn about this situation.
It is possible to represent yourself in a civil suit for $15,000.00. However, it is a little more complicated than simply filing some forms (and expect the airline to vigorously defend the action - I would recommend carefully reading your ticket agreement and see if there is any cap on damages placed in there, a lot of airlines and other travel companies do place limits on how much they are liable for in these contracts!).
Here is a general overview of civil litigation in Florida (from the FL Bar Association): https://www.floridabar.org/DIVCOM/PI/RHandbook01.nsf/1119bd38ae090a748525676f0053b606/966753dab01ea462852569cb004ca402!OpenDocument
To start, the first documents you need are a civil complaint and a summons.
The Complaint is the first document you file (in your case you would be filing a complaint for "breach of contract" and "negligence"). You can find examples in your local law library (along with a lot of helpful information and resources for how to prosecute your claim and the steps for civil litigation).
The other document, the Summons, is issued by the court when you file your complaint (bring extra copies with you). It tells the defendant that the action has been filed, where the court is located, and what the first hearing date is.
Take the two documents (the summons and complaint) to a civil process server (you can contact your local bar association for referrals) and the process server will "serve" the airline with these documents, then give you a "proof of service" to file with the court, and your lawsuit will be initiated.
Short of filing a lawsuit, you can try to mediate the dispute with them - contact your local bar association and request referrals to mediators, a third party neutral can often help you reach a mutually agreeable resolution. Use the bar association's referrals to contact a mediator or two, the mediator will then contact the other party to set up a mediation session, and you can go from there - hopefully resulting in a formal or written settlement agreement, and save yourself the time and expense of litigation.