I don't think @Reynard is asking for advice on how to increase immersion by increasing narrations of bathtime. Apart from anything else, you can immerse just as easily in the look and smell and sound of the dragon you're slaying, so bathing has nothing special to offer on this front.The second part, immersion, is when a player desires to live a second life. They don't want to play D&D, they want to live it. Therefore, mundane stuff is welcome to them. They want to describe their baths and how they eat and experience a world that reacts to those small details.
In the case that players don't want to engage with that, fine. Let them not be engaged. That's how they have fun. However, it would be unfortunate if the DM never gave the players a chance to try the more immersive side. That's why I ask about the finer details, even if they didn't have anything special prepared for the scenario.
Reynard seems to be asking how to make the fiction of the PCs' lives more closely resemble real human lives. I think the answer is therefore to look at how your game generates fiction and what sorts of fiction matter to it. And this doesn't have to be done in the abstract. There are actually many, many easily-available RPGs that have solved the problem Reynard describes.