If a general contractor did shoddy work in my kitchen and I just found out that despite what his website represents he is in fact not licensed by the state, what is my recourse?

You can sue the contractor for breach of contract and fraud.  Breach of contract would be for the poor quality work.  Fraud would be based on intentional misrepresentation that the contractor was licensed.  You would file one lawsuit in which breach of contract and fraud are separate causes of action claims.
Your damages monetary compensation for breach of contract would be a partial refund for those items of work that were of substandard quality.  Other work done to acceptable standards would not warrant a refund.  An alternative to seeking a partial refund would be to hire another contractor to complete the work.  Your damages for breach of contract would then be the cost of completion.  If you hire another contractor to complete the work, you will need to mitigate minimize damages by hiring someone whose fees are comparable to other contractors in the area.  If you were to select the most expensive contractor you could find, you have failed to mitigate damages and your damages will be reduced accordingly.
Damages for fraud include punitive damages which are a substantial amount to punish intentional wrongdoing.
As for preventing the contractor from defrauding others in the future, there is not much you can do individually beyond what you have done in contacting the Florida licensing board and having it take action against the contractor.