Conditions pertaining to cancellation of a contract are contained in the contract, that is there is no general "right to cancel a contract", apart from certain 3-day window rights in some jurisdictions. For example, Washington state has some specific cancellation laws, summarized here. If the contract does not allow for cancellation, then you can't cancel the contract. I assume that your contract does have a cancellation clause: then you could have to see what that clause actually states.
As it turns out, Florida has a law (Fla Stat. 501.017) which regulates health studio (gym) contracts, requiring certain clauses above the signature box. One required clause per (1)(b)2 is
that notice of intent to cancel by the buyer shall be given in writing
to the health studio.
If the health studio wishes to enforce the contract after receipt of
the notice, it may request the department to determine the sufficiency
of the notice.
This means that if they want to enforce the contract after a cancellation, they may request "the department" (Department of Agriculture and Consumer Services) to rule whether the notice was according to the contract (as mandated by law).
It will also contain under (1)(d)
A provision for the cancellation of the contract if the buyer dies or
becomes physically unable to avail himself or herself of a substantial
portion of those services which he or she used from the commencement
of the contract until the time of disability, with refund of funds
paid or accepted in payment of the contract in an amount computed by
dividing the contract price by the number of weeks in the contract
term and multiplying the result by the number of weeks remaining in
the contract term. The contract may require a buyer or the buyer’s
estate seeking relief under this paragraph to provide proof of
disability or death. A physical disability sufficient to warrant
cancellation of the contract by the buyer is established if the buyer
furnishes to the health studio a certification of such disability by a
physician licensed under chapter 458, chapter 459, chapter 460, or
chapter 461 to the extent the diagnosis or treatment of the disability
is within the physician’s scope of practice. A refund shall be issued
within 30 days after receipt of the notice of cancellation made
pursuant to this paragraph.
It appears, then, that you simply need to give notice in the correct form (written, with a doctor's notice). In case the club does not have the required provisions in the contract, the contract violates state law and is unenforceable.