Responsibility
"The one who cares the most will be the one to change." The kid who is peeing needs to have the most care and responsibility about the pee and clean up. The parent's job is to provide cleaning supplies, time, and a hug.
Ages 3 - 5: Children this age need some help and encouragement with the clean-up process. They should do 75+% of the work, with you helping with the big/heavy parts and giving encouragement.
I usually give fill the tub/bucket for them at this age, add the vinegar, and help them wring out the bigger bedding (after they have asked respectfully!). I will also help them load the washer and dryer, and may help with folding the bigger items if they're really tiny (again, I only help after they've asked, and I let them know they can do that for certain parts).
Ages 6 - 13: In this age range the child does 98.5% of the work. I will add in the vinegar, start the washer and add the soap, and start the dryer. They haul, wash, scrub, wring, load, switch, fold, et cetera, et cetera.
Ages 14+: Usually
by this age they are strong enough to start the washer and dryer on their own and add vinegar and laundry soap. I will usually tell them to let me know when they're done (load is in the washer and washer started) and I will check the area to make sure puddles are cleaned up, etc.
Pro-active Ideas
Pee-ce mat
Get piece of good, rubber-backed outdoor carpeting to put at your child's pee spot - they're great floor protectors! You can often get sample pieces or discarded scraps from hardware or flooring stores.
Waterproof mattress
It may be worth investing in a waterproof mattress if your child has ruined a good mattress already. You can find them online through medical or campground supply places.
The Pee Kit
Get a mop bucket and load it with a bottle of vinegar, a scrub brush, and several rags. When your child pees they will have everything they need to clean up. Keep these handy - some families have one or two in the house and one in the vehicle. (If you're feeling creative and have a bit of extra energy (!) decorate the bucket for your child; let them know that you are OK with their hobby and choices and love them anyway).
Urination Station
This is the spot where your kiddo can scrub out their clothing/bedding; it can be the tub, laundry room, sink, or outside on the deck (great in nice weather, fresh air is very beneficial! (also keeps the pee smell out of your house - bonus!)).
Keep a pee kit near this spot so your child has everything they need to clean up. Putting a pee-ce mat near this area is also a good way to protect your floor when they ready sit.
Mattress protectors