Ltd Company in USA
To register the UK company in Florida, you would first need to obtain an equivalent of a certificate of good standing certified with an Apostille in the UK, then you would submit those documents to Florida Secretary of State as part of foreign entity registration.
Another option you could look into would be registering an LLC in the state of Florida to hold the property. You could list your UK company as the owner of the Florida LLC, and even register the LLC under the same name as your UK company, provided it was available in the state.
While the second option would be much simpler and require less costs and process time, it would be a very good idea to consult your business plans with an accountant.