When you land at JFK, you're going to have to clear Immigration, then collect your bags and clear Customs.
If I were you, I would book a room at one of the hotels near the airport (look for one with an airport shuttle, and preferably an onsite restaurant.) Call the hotel from the courtesy phone inside and let them know you're ready to be picked up. Take the train that connects the terminals to the hotel shuttle stop and go try to get some sleep - you need to be back at the airport early in the morning (I aim to have 2 hours before departure - JFK is busy and I'd rather check in, check my bags, get through security and use the extra time to find breakfast in the terminal.)
Once you factor Immigration, Customs, travel to and from a hotel, a meal, and two hours to check in next morning, your layover has shrunk to about 9-10 hours, just enough time for some sleep and a hot shower.
Hope that helps - Enjoy Florida!

Nowadays all you need is the confirmation number (aka Record Locator Number) and your I.D. (in your case your passport since you are traveling internationally) rather than a "real ticket" in order to fly. When you get to the airport you simply check -in at the "ticket counter" and get your Boarding Pass. Depending on your airline you will probably even be able to print your Boarding Pass "on-line".

Now, you'll be arriving at JFK around 5PM and you'll have to clear Immigration and Customs. Once you exit the Customs area you'll be outside the secure area and while there are a few "shops" outside the secure area there are not near as many as there are inside the secure area so hanging around the airport is not really your best option. In general, the areas outside the "secure area" are not the most comfortable for "waiting and passing time.

Also, with a 15 hour layover and your flight to Pensacola being the next morning you won't be able to recheck your bags until the following morning so you'll be lugging your bags around with you while you "browse". Keep in mind also, that if you won't be able to bring any "checked luggage" past the TSA checkpoints so unless you are doing "only carry-on" you won't be able to get to any of the shops/restaurants in the secure area until the following morning.

I note you asked "Where should we sleep to save some money?" - well, whatever you do don't even consider catching a few zzzz's in the "waiting areas" inside the airport unless one of your party remains awake at all times otherwise you run the risk of loosing some or all f your belongings.

So, do as txgirlinbda recommends, get a room at one of the many nearby "airport" hotels (you'll find them in all price ranges) and have dinner, relax and get a decent night's sleep before going back to the airport the allowing morning for your flight to Pensacola.

The airport hotels at JFK aren't that great, but I don't think you have time to get one close to Manhattan since you have to be back at the airport so early. With this much time, I'd ordinarily say go into Manhattan and walk around, but it's likely you wouldn't get there until after 8, and I suspect you'll be exhausted coming from Kiev. Food is likely to be terrible at the airport hotels, but again, you probably don't have many options. Expect to pay no less than $150 for the overnight stay.