Like the other poster below, I've written code for electromechanical automation long enough to have some idea of what you're in for.
First consider cost effectiveness: you can hire someone for $30/hour to clean your house. That covers the bathroom and probably another room or two. Nothing you're likely to build will beat that for a few years.
Second: you're looking at it from the "wrong" perspective. Instead of building a robot to clean the bathroom in the way you specified, it's probably easier to build a bathroom that's designed to be cleaned automatically. Imagine towel racks that rotate into the walls to keep them safe, vacuum-augmented drains in the floor, water proof floors and walls and a rotating wash head that can descend from the ceiling dispensing high pressure hot water and detergent. Combine that with high volume air vents, and you can clean the room in less than a minute.
Now, if we venture back into the realm of feasibility, why not a version of the Roomba that only does the floor, but has a scrubber, wet vacuum and washes with hot water and soap. Floor's usually the dirtiest area anyway and you can build that kind of cleaner for a few $100.
A maid would still do the job better and cheaper, though.