By the plain wording of Amendment XIV, Congress' power is extremely broad. Broader than it should be, most likely, but if someone is being put to death without having committed any crime, that would be a violation of Amendment XIV; if Congress finds that such an action is taking place, XIV.5 would seem to empower them to pass legislation to stop it.
Under Supreme Court precedents, some of them recent, Congress can pass a law under the enforcement clause of the 14th Amendment only in response to a pattern of constitutional violations, not a single case. Maybe the Supreme Court is interpreting the enforcement clause wrongly here, but anyway that's the current law.
It's possible to find such a pattern, but that would only support a law of general applicability that goes to a class of disabled people, not a bill targeted to one case like the one here.
Congress may also constitutioanlly alter the jurisdiction of the federal courts to give in effect habeas review of this kind of case to the federal courts; again, as applied to a class of cases, not to 1 person.
I don't think the law to be passed at 12:01 this morning will ultimately hold up, because it's a bill of attainder. However, it will hopefully be an effective stop-gap measure until something better is passed by Congress or in Florida.
My last point is that the pompous windbags in the Senate always screw things up. The did during Impeachment, and they did here. Tom Delay and the House wanted to pass a general law, but the Senate wouldn't go along.