How to get smells out of a foam mattress?
Your local pet store will carry a product called Nature’s Miracle, which comes in a white bottle with red print. The stuff really works. Though you probably want to get as much of the urine out as possible first, either with towel or a wetvac or a combination of the two. I would suggest renting a carpet cleaner that would inject clean water and pull it and hate urine out, but I don’t know if that would work with a tempur-petic mattress.
In the interest of actually answering the question that was asked, I don’t see any way you could clean this without introducing more liquid into the foam. You could try putting some kind of cleaning liquid into a device like a clothes steamer or humidifier to turn it into a mist, but any cleaner strong enough to dismantle cat pee molecules is also likely to wreak havoc on any nearby lungs once airborne. I think the way to go is several small, light treatments over time. Put a little bit on, squish it through some, let it sit for a week, then repeat as needed. I’d favor Nature’s Miracle. It has no real smell of its own, and it’s worked for me on all but the worst cat pee stains. Bleach sounds like a disaster in the making. This stuff isn’t even supposed to get water on it — how do you suppose it’ll hold up to having acid poured on it?
I would strongly recommend the Nature’s Miracle (which I’ve even seen in non-specialty places, such as WalMart). It’s likely to be much gentler to the mattress, compared to bleach or vinegar solutions. I’ve used it myself on a futon mattress; worked like a charm, though you do need to apply it liberally to make sure it gets to all the places the urine did.
Plain ol’ cheap white vinegar. It neutralizes the cat odor smell. Use 1 part water to 3 parts vinegar, or if you really hate vinegar smell, 1 part water to 1 part vinegar. Dampen the mattress but don’t soak it too thouroughly. You may need multiple applications. Yes, it’ll stink even worse at first, but in a few days all odors will be gone completely. I was in a similar situation a few months ago, and vinegar worked where no other odor neutralizer would.