You got a complaint from a customer.
Write a letter to the customer. In your letter
1) describe why you made the mistake
2) tell the customer how you would make up for this
I received your letter yesterday complaining that the central heating system in your new house had not been completed by the date promised. I feel awfully sorry that the problem has been caused by my mistake about the date for completion on referring to our earlier correspondence.
I mistook your date of completion for next week due to my inattention, so the working team installed the system according to the wrong date. I realize the inconvenience that our oversight must be causing you and will do everything possible to avoid any further delay. I have already given instructions for the work to have priority and the engineers working on the job to be placed on overtime.
I assure you that these arrangements should see the installation completed by next weekend. I hope the settlement of this matter can meet your wishes.
The fault is entirely mine and I deeply regret that it happened.