Unfortunately it does sound like you may have killed off the beneficial bacteria needed to transform his toxic waste products into harmless ones. Letting a tank sit for 24 hours is not cycling though I am aware that this is the advice given in some pet stores and tank pamphlets. A cycle can take weeks to establish unless you use a products specifically meant for seeding your media with the beneficil bacteria. Some products are good and some are a waste of money, best to read up before buying.
With a betta though you can safely get away with a fish-in cycle because they don't need to use their gills for all of their oxygen. Plus they are a low-waste fish so probably won't cause an ammonia spike so high it will cause damage before you can remedy the situation. Get a liquid test kit if you can, not the strips, they're inaccurate. Test your ammonia, I expect it's high. Keep doing small (~25%) water changes to keep the ammonia around 1.0ppm and use something like Prime to render the ammonia that's left harmless to your betta. You need to have some ammonia for the bacteria you want to grow in your filter!
When you start seeing nitrites on your test you'll know you have the first wave of bacteria you need. Keep doing the water changes because nitrites are toxic too! Bacteria will move in that will start eating the nitrites and converting them into nitrates. Nitrates are much less toxic and are actually usable by plants. Monitor them though and keep doing your water changes. I actually lost a betta to a nitrate spike once. Well, actually not the nitrate spike but my reaction to it. I panicked and did a massive water change to bring it down not realizing that such a drop was as dangerous as a diver returning from deep sea too quickly. He got the bends and died
One of the symptoms of too-high nitrates is the betta lying on the bottom.
Good luck!