Hi, Sue! Thanks for responding!.
There are a betta log, several artificial plants, and some live plants. There is an airstone going at all times. The neons were happy, pretty much, for a year until the hood light went out and I replaced it with a LED one. I have angled it back and over one corner of the tank, as it is much brighter than the other one was - so the effect is as if they are in a stream and come out of the overhanging brush into a beam of sunlight. They don't seem to care one way or other about it, though. They neither avoid nor group there in the light. They are not swimming together as a little school, but each is doing his/her own thing and occasionally a few of them will hang out together. So that might just be a coincidence of timing.
The tank is 45cm x 25cm x 21 cm and holds 5 US gallons (about 19 litres). A 20 gallon tank is on our wish-list, definitely. We want a tank that's taller than the usual footprint, and plan to plant it heavily. But that's in the future.
Kanaplex is kanamycin sulfate, supposed to be very good for dealing with a wide range of bacterial (gram pos and gram neg) infections. Not doing much for my little guys, though.
Amonia/Nitrite/Nitrate are all ok ok. Tap water is on the soft end of the scale, per the local water board. Specifically, they say:
Calcium mgCa/l 5.60
Magnesium mgMg/l 0.73
Hardness as mg/l CaCo3 16.95
I'm wondering about the corys spawning.... Seriously, I have never seen one female lay that many eggs, and I've had corys for years and years. There were hundreds of eggs. And then in a matter of a few days there were none left. I had assumed they'd been eaten, as I haven't seen any fry. I wonder if they've done something to the water quality that is not showing up on my test strips? I didn't vaccuum the gravel really well last week, as I was afraid that there might be some fry there that I couldn't see. I'll have to give it a thorough job tomorrow morning, and hope that if they are there they will find some other part of the tank to be in.
The only other treatment I have in the house is Tetra Medica Fungistop Plus. I did try that early on, but it didn't seem to do much which is why I thought we were dealing with a bacterial infection. It can't be used in water that's had water treatments to remove heavy metals, which I think is something that's in most of the water conditioners I have. I'm wondering if it actually is a fungus, and not columnaris bacteria, after all? And it's weakened the fish enough that a secondary bacterial infection is now running wild? I keep the water temp at 76F (about 24.4C), which is what they have always seemed to like.
I guess it'll be another water change in the morning and then I'll try Fungistop one more time.

And will check out the two you recommened.