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
Rebuilding the kitchen will not stop them. Various insecticides will kill them for now. Borax is safe and effective long term . When I built my house in roach country ( E. TX) I put several pounds of borax in the walls before dry wall was put up. Worked great. After 25 years roach free years , apparently a couple recently got in and found the dishwasher cabinet was ideal ( no borax nearby). I was seeing a big roach about every other day , after none for years. I drilled a 1/2 hole under a counter top and blew in borax powder around the dishwasher. In the last month there have been only a few little roaches and they are slow and easy to kill because the borax slows them down before they die. So you can have an exterminator every few months or figure how to get borax powder where they are living. ( My wife has not noticed the hole so I am good). ADDENDA : On the otherhand ,if you are going to redo the kitchen anyway, put borax under, behind, and everywhere else around new fixtures, that may fix the roaches .