You might be able to do it at that price for everything but the toilet. Instead of going big, the design I would use (after only thinking about it for 2-3 minutes) would be to mimic an insect. Go really small and have it be able to adhere to the surface and clean as it went (tub, glass, floor, etc). Use a laser/camera to analyze the surface. The key would be to keep the center of gravity as close to the wall as possible.

On the large side, I would go with a pivoting telescopic arm with variable cleaning instruments. Use a counter weight to balance the robot.