Have you got a clear view of why the standard of cleaning is not good enough for the agency? You are going to need this knowledge... You need a written list ( email) from your agency as the first thing or you won't have a leg to stand on.
Then you need to look at the contract you had with your cleaners. The things that the agency say are not properly clean.... Did your original agreement or contract with your cleaners cover them, and to what standard?
When you have a clear view of this, you will be in a much better position to proceed.
It doesn't sound likely that your letting agent would wait for a reclean, though this would be my first choice. And it doesn't sound like your cleaner wants to play fair. But if you can get your cleaner to do a reclean and your letting agent to specify exactly what needs cleaning and how, then do that.
If not, then you need to look at the discount route. A discount can be the full original price in appropriate circumstances. If you are in the position of having to pay new cleaners, plus pay for damages, I'd say that's a pretty good reason for a discount in the amount of the full original price.
Tell them that, and tell them you are happy to go to small claims court about it, but wahtever you do, don't pay them!