I agree with Laura's Mom. He'll be just finel It is very, very common for puppies to get worms. What type of worms did the vet say he had? I'm sure the treatment he/she gave you applies to the certain type of worms. Do you have any other pets that these worms could be passed to? I heard that a cat or dog sniffing the others butt could pass them the worms as simple as that but I don't know how much truth there is to that. I'm sure your vet gave treatment to fight the specific type of worms. Does your dog eat poop? his own or other dog's poop? This can cause worms or if he eats his own it could cause reinfection.
Here are some natural remedies for dog worms that may help compliment the deworming treatment your vet gave you and prevent worms in the future;
Garlic - Garlic is mostly used due to its medicinal properties. You can moderately give garlic to your dog. This will help in the internal parasite expulsion, mostly the intestinal worms. Garlic also increases the immunity and reduces the dog’s cholesterol level.
Wormwood - Wormwood has calmative properties and is a digestive tonic. Feeding wormwood to your dog will help in the worm expulsion.
Fennel - Fennel helps in the intestinal worm expulsion. It also helps in improving the immune system.
Clove - Clove is a strong antiparasitic, which helps in detoxifying the dog worms.
Pumpkin Seeds - Tapeworm infestation in dogs is mostly treated by feeding freshly ground white pumpkin seeds.
Coconut oil.
There are also herbal remeies they sell commercially that may be safer than conventional dewormers such as the ones linked below;
Natural Worm Treatment
Molly's Herbals - Herbal Wormer
All Natural Dog Wormer | Canine Deworming Herbs
As far as killing the worms/cleaning the area where your dog went in the house I'd make sure to wash the area well with soap and water or your cleaning product of choice, but so ensure safety of your pet and to be sure and kill the worms/eggs you may not be able to see wash the area with hydrogen paroxide.
I found this online;
You can kill parasites on household surfaces with hydrogen peroxide. You should also know that hydrogen peroxide is cheap, easily available and easy to work with. Hydrogen peroxide has been a household staple for decades, but you can add a new purpose to the list by using it as a general household sanitizer. Although medical experts no longer recommend using hydrogen peroxide to disinfect cuts and scrapes, the chemical is safe for sanitizing kitchens and bathrooms.
Difficulty: Easy Instructions
Things You'll Need:
One bottle of 3 percent solution hydrogen peroxide
One spray bottle with opaque plastic reservoir (dark brown or black)
Clean rag or paper towel
Step 1 Clean a spray bottle that has a dark brown or black plastic reservoir on it. Fill the bottle half-full with tap or distilled water.
Step 2 Fill the rest of the bottle with hydrogen peroxide. Put the sprayer on tightly and shake to mix the solution.
Step 3 Spray the contaminated surface with the hydrogen peroxide solution. Allow it to soak for 60 seconds.
Step 4 Wipe up the solution with a clean rag or paper towel.
Step 5 If the surface still is contaminated, clean it again. You may need to use commercial cleaners or scrubbing agents to remove gross contamination before attempting to disinfect with the peroxide solution.
Also it is important to wipe/wash the anal area/feet of your dog a few time per day to prevent contamination/spreading of worms.
And as Laura's Mom said, take another stool sample into the vet to see if the worms are gone and also keep a look at the poop to see if you still see worms. But just because you don't see them doesn't mean they aren't there. Some are very small and difficualt to see with the naked eye and there also may be aggs.
Good luck!