kind cooperation

[Unregistered]

Is this a correct sentence. I'm not sure if kind can be there before cooperation.

we ask for your kind cooperation

[Anglika]

It is a correct sentence.

Whether it is the correct sentence in the context of use is another matter.

[David L.]

"kind cooperation" is now a standard phrase (collocation).
"We ask for your cooperation" is a little 'cold', brusque, formal. The request is 'softened' by adding the 'kind'.
'kind' means 'having or showing a friendly, generous, and considerate nature'. So, if 'ask for your cooperation' seems like bossing someone/people around, then 'kind cooperation' has the sense, 'please be considerate and understand that there must be a good reason why I am asking you to do this.'

It is now just a polite way of making a request.