At first, I was skeptical that this would work in terms of washing the clothes properly, but the more I think about it, the more it seems that it could and that it could be developed into a really cool product.
What it would take is to build such a system as sort of a game, it could be real fun - to wash the clothes properly you'd have to follow the instructions on the monitor carefully, you'd first have to pedal slowly at a precise speed, to rinse the clothes, then maybe stop at certain times, pedal harder and longer at the washing phase and then finally start pedaling fast and constantly for the system to work as a centrifuge to squeeze the water out of the washed clothes.
I think even the speed and power required for the centrifuge function could be achieved by manually pedaling the bike through the use of various adaptive gears, if not, then the machine could be a hybrid using both manual power from the pedals and the stored electrical power generated by induction or simply taking electricity from the public network.
The machine would show you how you're doing during and after the process, to what extent you're reaching the requirements, and when you have to try harder. The cleanliness of the clothes washed in such a way would directly depend on your game score, so you'd have a motivation to improve every time. To do well it would take not only physical fitness but also the ability to carefully follow the instructions which would train your mental performance and movement coordination as well.
Just like in the case of ordinary washing machines, there would be different programs intended for different types of clothes and different types of washing. They would require different physical strength and mental performance accordingly.