I would like to kindly ask you could you check......?

Too complicated and long-winded is one thing, but it also doesn't sound right - I think it may be because you're repeating "you" twice, though I'm not sure. Try one of these:

  I would like to kindly ask you to check ...
  Would you be so kind as to check ...
  Could I ask you kindly to check ...
  Could I kindly ask you to check ...
  Could you please check ...

If, on the other hand, you want to ask whether somebody has already checked your file, you should go with:

Would you be so kind as to tell me if/whether you have checked …
Could you please let me know if/whether you have checked …
I would [greatly] appreciate it if you could let me know if/whether/when you have checked …

The problem with to kindly ask (apart from the split infinitive, which is cumbersome) is that it says that you are being kind when asking: you are asking kindly. It's usual to ask for someone else to be kind and do something.

I would like to ask [you] could you kindly check......?

However, "I would like to ask" is meaningless, because you are asking. "I would like to ask" is more usefully followed by "but I won't". It's far more likely that the question would be

Could you kindly check ... for me, please?

The structure to asking a question usually goes as [verb] [object].

Do you know the time?

If you would like to be polite, you'd say "please."

Please check . . .