Sounds like the puppy has way too much freedom having the entire kitchen. Instead of giving him the kitchen, how about crate training him? He'd need a crate that isn't so large that he can pee on one end and sleep on the other. They don't like to soil their "dens".

I'd have him on a schedule of taking him out every 2 hours during the day. And crated at night, and have the crate in the bedroom, so that you can hear him when he whimpers that he needs to go out, so that you can get up and take him out immediately. He needs to learn to be able to trust that you'll take him out when he needs to go.

When he's out of the crate, he must be on a leash at all times, attached to you. He has no opportunity to fail this way. You will see him start to circle to go potty and can take him outside immediately.

I baby sat a 10 week old schnauzer puppy for a week for a friend, and we didn't have a single accident in the house by following this method.