You cannot say:
"Laquelle de ces bouilloires marchent la mieux" , it is, le mieux , because of the verb 'marcher':
'le mieux' is an adverb as it modifies the verb 'marcher'.
You might say/ hear:
C'est la chanteuse la mieux payée au monde entier = She is the best paid singer in the whole world
C'est la banque la mieux représentée en Angleterre = This is the bank best represented in England
as here, la mieux
modifies the adjective and can agree with the noun although some purists would dispute this and say that it is better French to use 'le mieux' also in both those examples
But you will hear both...
Hope this helps!