I'm going to be long-winded here so bear with me.
One of the problems with Bettas is that they are kept in overcrowded conditions even before they are shipped to the stores. Another problem is that they are often kept in those useless plastic cups where the fish just sits in its own waste. Think about it, this would be like locking your dog and cat in a closet and letting it live in it's own urine and feces.
Back to then cups/vases/etc. When kept in cramped conditions bettas won't be able to keep and maintain their muscle tissue and as a result they will live a shortened life and die from a fatty degeneration of their tissue and muscle atrophy (which is basically a loss of the muscle fiber).
Another thing that doesn't help is that bettas are inbred and overbred commercially that their genetics aren't what they should be, much like many of the common domestic livebearer strains. Bettas are prone to tumors, bacterial infections such as Flexibacter columnaris
, finrot and dropsy.
Based on my own experience I feel that a 5 gallon should be the absolute minimum tank that a betta should be kept in. There are heater made to fit small tanks of this size. Personally, I feel that a 10 gallon is a better choice in that it's easier to heat and it also gives the betta more room to swim. This tank also gives options for a couple of tankmates such as Cherry barbs, Corydoras catfish, Kuhli loaches or Harlequin rasboras. Also, the bigger the tank the more forgiving it is of mistakes. That said, there isn't much room for error in a 10 gallon but the margin is still larger than it would be for a 5 gallon.
Bettas will do best in temps between 75-80F. Any lower than 70F and they tend to become lethargic and will only make movements that are absolutely necessary to it's survival- such as eating- and will lay on the bottom. The opposite is true of temps over 80F. The metabolic rate of the fish will be increased causing the fish to become very active but this increase in metabolism will shorten the lifespan of the fish.
If the tank that your fish is in isn't heated this could be the reason why it's laying on the bottom and not moving. My initial thought was a swimbladder disorder but, like the others already mentioned, it could be anything. I was over my parent's house a couple of years ago and Dad was doing a water change on his 29 gallon tank (which I now have) and refilled the tank with cold water. I noticed this because his gorgeous male Pearl Gourami was laying on its side at the bottom of the tank. I checked the water and it was cold so I started doing some rapid partial water changes will warm water to bring the temp back up. As the temp approached what it usually was in the tank (78F) the fish started to stir and came around. He ended up being okay.
Filtration is fine in a betta tank as long as the current is very light. Powerheads and mechanical filters that create a lot of turbulence should be avoided. In a tank with too much current the betta will become stressed and the constant current can wear down the slime coating of the fish leaving him/her susceptible to bacterial and other types of diseases such as Ich. IMO, a small air-driven sponge filter is the way to go. The air flow can be regulated by a valve to help keep the turbulence low.
A betta will also benefit from having live plants in the tank. Since a 10 gallon tank doesn't throw off a lot of light, low-light plants like Java ferns, Java moss and the floating plant, Hornwort, all make great plants to use for a betta tank. Not only do the live plants provide cover to help the fish feel more secure but they also aid in the biological filtration of the tank by taking in ammonia, nitrite and nitrate.
Since you have only had the fish for a couple of days the tank will have to undergo what is called cycling. I can rail on ad-naseum about ammonia, nitrite and nitrate and it's effects but short and sweet, here is a quick rundown on what the cycle is:
Cycling is the process of developing nitrifying bacteria in the filter and on every surface of the tank. This bacteria is what is responsible for the biofiltration of a tank. Once established, this bacteria will convert ammonia to nitrite. Another bacteria will convert nitrite to nitrate. Ammonia is highly toxic to fish. Nitrite is less toxic than ammonia while nitrate is the least toxic of the three compounds. Normally, with a regular maintenance schedule, nitrates can be kept around 10ppm without much difficulty.
Here is what you need to know about cycling:
The cycle is started when fish are added to a tank. Fish give off ammonia from their gills through respiration. Fish also pass a dilute ammonia in their urine. Leftover food, decaying plant matter and other waste products will also give off ammonia. All ammonia is toxic at .25ppm or above. The amount of ammonia that it takes before a fish succumbs depends on the species. In a cycled tank there will be 0 ammonia.
It can take up to two weeks before enough nitrifying bacteria is established to bring the ammonia down to 0. Once this biofilter is established, it will remain in the tank and grow and decrease as you add and lose fish.
The bacteria responsible for the biofiltration is aerobic. This means that it loves and needs oxygen to survive. The majority of biofiltration in a tank takes place in the filter. This is why it is important to have a good quality filter with lots of room for sponges or filter floss. This will give the bacteria a larger surface area to colonize. The bacteria will also live on every surface inside the tank as well. They can't be uprooted by gravel vacuuming. The following is what can kill the bacteria:
1.Chlorine- never rinse your filter media in tap water. This will kill off the bacteria colonies. Rinse the filter cartridge, sponge, floss or whatever you are using in a bucket of used tank water and reuse it.
2. pH swings
3. poor filtration
4. temperature changes
5. daylight/light- this why filters are dark
6. medications such as antibiotics (regardless of what the packaging says)
After the initial two week period you will see a climb in nitrite. While nitrite isn't as toxic as ammonia it is still toxic-usually at 2ppm or above. To counter nitrite toxicity, add 1 teaspoon of aquarium salt per 20 gallons. Table salt is fine to use but it must be non-iodized. Water changes and gravel vacs must also be done to remove high nitrite levels. Salt isn’t and never should be considered a replacement for water changes.
Nitrites will remain very high for about six weeks. The bacteria responsible for converting nitrite is a painfully slow growing one. These bacteria are also sensitive to the things listed above.
One day you will find accumulating nitrate with falling nitrite. When the ammonia and nitrites both have a reading of 0 and you have a noticeable buildup of nitrate the cycle is complete.
Once complete, a regular maintenance schedule of weekly 20-30% water changes should be done along with thorough gravel vacs to remove fish poo, leftover food and other waste. This should be enough to keep your nitrates between 10-15ppm. If your nitrates are higher then you need to cut back on the amount of food that is fed daily or increase the frequency of water changes. The lower the nitrates the more your fish will thrive.
Last but not least (for now) I want to touch on water conditioners. I don't know what type of water you have in Kansas (city, well water, etc.) but many communities have will add ammonia to the chlorine in the tap water which creates chloramine- a more stable chemical than chlorine that is also used to treat water. Water can also contain trace elements of heavy metals. I use a water conditioner that neutralizes both chlorine and chloramines as well as toxic heavy metals. It might be a good idea to get a water report so you can see what is in it. In rural areas it isn't uncommon to have a high level of nitrate in the tap water.
Hope this helps and give a shout if/when you need more help/info.