Word for repairing something in a way that changes its function

That's called repurposing.

Repurpose (verb): Adapt for use in a different purpose.

Example: They've taken a product that was originally designed for the CD-ROM and repurposed it for the Microsoft Network. [Lexico]

Example: You repurposed the washing machine for cloth-colourer.

If you fix a broken thing and make something else out of it, I suggest Upcycle.

• Transmute would also work well. It means to change something in form, nature, or substance

• Or use customise if you change it according to your needs.

• Revamp, remodel, transfigure and retool could also be used.

I had a bicycle and I changed it to a car.

For this example, I'd use transmogrify.

You've adapted your washing machine.

Adapt: to change, or to change something, to suit different conditions or uses. [Cambridge English dictionary]

Adapt: Make (something) suitable for a new use or purpose; modify. [Oxford Dictionaries]

How about modify or retrofit? From M-W:

modify: to make basic or fundamental changes in often to give a new orientation to or to serve a new end

retrofit: to adapt to a new purpose or need

Your example:

The washing machine was broken and I modified it in such a way that the function of washing machine is no longer washing clothes but, say, now it colours the clothes.

The washing machine was broken and I retrofitted it in such a way that the function of washing machine is no longer washing clothes but, say, now it colours the clothes.

I believe that in the current computer era, you can also use upgrade for that purpose. Upgrading is the act of replacing a program with a newer version of itself with better performance and new features besides the fixing of whatever errors the previous version had. It has become so common that the use has extended beyond the scope of the software.