Nickelback MP3s
I had a bunch of CDs (for data) I wanted to get rid of. I threw them in our fire pit, doused them with lighter fluid, and set them afire. They melted nicely.
That is pretty shitty thing to do. All kinds of materials in the CD that shouldn’t be getting burnt.
This is the key point.
Simply cut them - 5-inch with scissors, 3.5 with tin snips or some such. For 5-inch, cut in two or 3 places so it’s hard to find what goes with what. Use a magnet around the rim of the inside material. If you’re really fussy, MY work used to have a bulk paper slicer, you could put a stack of disks in there and reduce them to 1/4" strips bit by bit.
If the NSA wants to go through the trouble of reading then, go ahead. Years ago I had a head demagnetizer for reel-to-reel tapes, but it’s long since disappeared to who knows where. But for the average Joe, they gotta be really curious. Plus, recovery would be in fragments with a lot of analysis time needed.
For the 3.5" a heat gun should do the trick. Heat until it’s bending in half, like a surreal Dali melting clock, and nobody is going to read it. Then squeeze it with a pair of pliers to definitely deform everything. (IIRC serious heat will demagnetize too) Take a nail and hammer it through case and inside material at several points. Put them in a jar with a lot of acetone…?
I recently disposed of some really old laptops - took out the (small) hard drives, and crunched each side of the case with a big pair of wire cutters. Probably broke the platters too. If you have those big bolt cutters, even better.
Back in the day I was a big user of Zip discs. It’s been my experience that if left to their own devices, they will destroy themselves 
Yep. I had three experience the click of death. Out of like maybe six or seven I owned. Worthless POS tech.
Simplest - I believe the main directory for most removable disks is along the outer rim. Take a chisel and hammer a cut radially that cuts the outer rim of the magnetic material inside and the NSA may read it, but nobody else will bother.
The thing about the magnet to degauss them is that you’re not certain that the data can no longer be read. Disc forensic analysis might still be able to retrieve some or much of the data — depending on how thoroughly you’ve degaussed them which, how can you tell unless you perform that forensic analysis. Too much trouble and not enough confirmation for me.
See the company, AccessData Group, now Exterro.
https://www.linkedin.com/company/accessdata-group-inc.
To be certain, I would take a pair of tin snips and cut them in half. Then I’d throw them into a large fire and ensure they are melted.
Quick, easy, and easily confirmable. With those actions I’d be satisfied that they are completely ruined.
“burn until the ashes are white” is a line from the SOP manual we had for destroying floppies and tapes at my old job.
Try as you might, but information is never lost. 
Dry ice and 99% isopropyl alcohol makes a fierce cooling bath. Immerse disks individually; fracture with hammer.
This:
And this:
Seriously, you don’t need to do anything else.
Would a regular magnet really do much to a floppy? I don’t remember them being that sensitive.
Agreed. I would do the same with the floppies too – run a drill through the floppies in a stack. It only takes a few seconds and not much mess.
and if you’re still worried about the CIA recovering the data, don’t throw all the pieces away in one garbage can.Spread 'em around.
But if you throw them into a large black hole you can buy yourself some time.
Sagittarius A* should do the trick. It’s super-massive and in the galactic neighborhood.
But Sgr A* will evaporate in about 1087 years. The OP said the data is sensitive, so they may not want to risk it being retrievable so soon.
IANAE but I do believe that if only some holes were drilled into the disks, or if they were cut in half, that there still would be a way to read some remaining data on the disk fragments. That’s why I recommend burning them.
Added: imagine a vinyl record. If it had holes drilled or if it was cut in half you’d still be able to play fragments of the music on it. There might be enough to even identify the song.
Returning to the real world.
If I was given the OP’s task, I would shell the disks, getting out the actual recordable media, and burn the media. The recording media is going to be a very small amount to burn, and won’t cause any serious issues. The shells get to be recycled or otherwise sensibly disposed of.
My sister has a rather lovely artwork created by a friend of hers. It is a wall hanging wooden sheet covered with little metal squares. On closer inspection, it is made up of hundreds of the sliding covers from 3.5 inch floppies.
Freeze them till they reach Tg, then shatter them with a hammer and distribute the tiny shattered shards in various rubbish bins around town.