There are plenty of recommendations out there on the web, but I can't say how well they work. I think the lifespan of a gnat is something along the lines of 15 days, so you basically just have to remove their food source, and they'll be gone in 2 weeks.
Clean dishes quickly, keep everything clean, get rid of potted plants (they feed on the manure found in many types of soil that you might get for potted plants) or cover the top layer of soil with an inch of sand to prevent them from getting to the manure, look for any dead animal bodies (rats, lizards, birds, etc.)
I've also seen ideas for traps, which supposedly work pretty well, but just because you catch some, that doesn't necessarily mean that you'll get rid of their breeding grounds. The basic idea is that you take a cup and put 1" of vinegar and one drop of dish soap in it, then cover it with something that resembles a funnel (curl up a piece of paper into a cone shape and cut the tip off.) They'll be able to fly in because the funnel will guide them downward, but they won't be able to get out so long as you seal between the lip of the glass and the funnel (maybe fold it back down on the outside of the glass and secure it with rubber bands. Like this:
We have fought this problem in our rentals before with success using the following techniques.
What doesn't work: Foggers. We treated the rental for MONTHS with weekly applications of foggers - 3 or 4 anti bedbug foggers per floor - hundreds of dollars spent - and the tenants had to leave the building for half a day every week and then deal with the stink and the toxic residues left behind. Not fun, and didn't work at all because bedbugs DON'T hang out where the fumes can get to them.
So, what do you need to do?
First, buy insect resistant, zippered mattress covers. These are usually some kind of plastic. You wrap each mattress in its own cover and zip it shut tight, sealing the bedbugs within the mattress inside. You must take care not to rip these covers or the bugs can get out again. This is done during the treatment process and for some time beyond.
Second, and this is the big step - Treat your entire house with diatomaceous earth (DE). DE is a non toxic, non pesticide means of controlling pests. It's a crushed silica product that is effective in killing not via poison, but by cutting the outer membrane of small insects.
Unlike most dust/powder which is smooth from erosion, DE has microscopic sized jagged edges. Many types of insects have an external shell which is coated with a kind of mucous membrane which helps them retain water. DE slices that membrane to shreds, and the bugs lose all of their moisture to evaporation, so they die of dehydration, NOT poison.
You can buy food grade DE online. It's safe to use around the house, and you can even dust your pets with it (keep away from the face, of course). We have three dogs and when fleas get in the house, we dust the animals and the areas they sleep with DE.
Now bedbugs are not jumpers, so the key with dealing with them is getting the DE where they live and where they travel.
Where they live: Your furniture. Those mattress covers? Open em up and dust both sides of the mattress LIBERALLY with DE, then close it up again. The bedbugs will get coated in it as they crawl around looking for an escape.
Your sofa/chairs - if you can, dust the INSIDES of larger pieces of furniture - heavily. Also apply dust in all the cracks and crevices.
Remove the wall plates - use a dusting sprayer like this:
and spray that dust into all the electrical junction boxes - the idea is to force the dust into the walls where bedbugs travel. (They don't like open areas, they prefer tight spaces and corners.
Also put a ring of dust down around your bed posts where they touch the floor, along the edges of the walls. Dust your carpets with it to.
I personally bought my supplies at Dirtworks ( http://www.dirtworks.net/Diatomaceous-Earth.html ) - I bought 2 five pound bags of food grade DE (which was 1 bag too many, honestly), a few puffer bottles, the glimmer sprayer, and we did ONE heavy application all over the entire property (we even dusted the lawn with the sprayer) and never had a bedbug complaint again.
Best Answer
These small flies are very hard to keep out because they are so small, so I think looking for what is attracting them is a better long term solution. This is a guess, but I think that using bleach in the drain is probably masking the scent of what is truly attracting them. A small flying insect making it's way through a water-filled P-Trap under the sink is unlikely, and unless you also have a sewer smell coming from the sink, the P-trap is probably working correctly.
Look for dead portions of houseplants, fruit or veggies that have fallen behind a cabinet or the fridge or in the cracks next to the stove, etc. Clean, clean, clean. Pull out the stove and the fridge and clean. Empty out the cabinet under the sink and clean that very well. Pull out everything from the pantry and get rid of the expired stuff, and clean.
I'm in Texas as well and anytime we have had tiny little flies at our house or my mother in law's, its been from potatoes that were forgotten or something else that was leaking or rotting out of site that we didn't know about.