Good question. For some reason, a lot of German speakers seem to want to use 'kindly' a lot. I have the feeling it used to be taught in 'business English' courses. But in modern English, it almost always sounds very wrong to me.
I would go even farther than Jo and say that 'We kindly ask you' is wrong, at least, not at all idiomatic. It is probably not particularly kind of us to ask you; otherwise we wouldn't have to be especially polite about it, would we? And we are not asking in a particularly kind way or manner; that doesn't even really make sense, unless we want to suggest that we usually ask people meanly or rudely.
The real point is that we want you to be kind and do as we ask. To my ears, the grammatically and logically correct way to use the word 'kind' in this context is
Would you be so kind as to (do sth.)?
(I've put the word 'as' in boldface because it's a mistake to leave it out.)
However, that phrasing is usually less polite, because it's so very formal and stiff. It's really more like 'Würden Sie gefälligst ... ' It could be polite, if you were speaking to a person and it was clear from your tone of voice that you meant it in a nice way, but it's not a phrase you would normally use in a letter.
So how do you simply ask someone politely to do something, in a normal way? I agree with Jo: the word you need is 'Please' (and at the end, 'Thank you.') If you want to be more polite, add a phrase with the word 'appreciate':
Please return the signed forms as soon as possible. Thank you.
We would appreciate it if you would/could return the signed forms as soon as possible. Thank you very much (for your attention to this matter).
(I've put the word 'it' in boldface because it's a mistake to leave it out.)
Hope that helps. (-: