Velvet can be hard to get rid of fully. Took my fish 2-3 weeks to recover from it.
One word of advice, do not stop medicating even if it looks like the parasites have gone away. This is why people often post about their problems with recurring ich and velvet. They both have complicated life-cycles and can only be killed at a certain point. If you stop your treatment early, you often don't kill it off and it will come back to plague your fish again and again.
I assume it will be the fluctuating temperatures that caused it. That's what happened with me. Left a heater unplugged in winter, the fish in the tank got velvet and because I shared equipment all my tanks and fish got sick.
You really need a heater to prevent the fluctuations in temperature that will stress a betta and leave them open to these opportunistic parasites. Also, the colder the water, the longer the life cycle of velvet/ich and so the longer it takes to treat.
I used ichonex when treating my fish. It was a copper sulphate/malachite green (I think) solution. Malachite green is light sensitive and will not work if exposed to bright light, so this is also why it is recommended to completely cover your tank if it is in the medication you are using.
Also be careful that he is does not scratch himself hard enough to cause ulcers. A couple of my fish cut their heads right open because the parasite caused so much irritation they would rub themselves raw.
Good luck with treatment. It is no walk in the park!