Nine times out of ten, if not more!, they last company lost the contract due to rubbish cleaners and poor management. Its ironic that you will have to keep the old cleaner there.
You'll have to tell them you have now got the contract and your standard of cleaning is a lot higher than the last company and you are more 'hands on' so will be keeping an eye on them. In my experience once rubbish cleaners know you are onto them they leave anyway.