From your examples, the best one would be:
The washing machine is broken, could you repair it?
repair refers to some actions that bring some kind of complex equipment from not working to working. Such repairs are often performed by persons with specialized training (such as an auto mechanic who repairs your car).
The washing machine is broken, could you fix it?
You could also use this, but fix has a more general usage for most anything. For example: "I will fix the water faucet." This could be asked of most anyone, for example your neighbor, regardless of any experience in the matter.
As already mentioned, "mend" in the US and at least my experience is only used when speaking of clothing or other materials related actions.