There are some ways around TUPE but one of them is not to cancel the contract for a day, it states in the TUPE legislation words to the effect of letting someone go under the pretense of not undertaking that role anymore but then hiring for the same role, which is exactly what that would be doing. You have to be careful, therefore it is essential to get the expert help that is required if you are planning on releasing someone. If you open yourself up to Tribunal for Unfair Dismissal it can be expensive.
If the cleaner in place is the owner of the business that provides the cleaning then chances are you do not have to take them on. If the cleaner has other employment with the company (some say 50% other say you should TUPE unless it is 70% on other sites) then you may not have to take them on. There length of employment on that site has some to do with it also. When they transfer it is as if they have worked for you the whole time. So if they have under a year then they have not yet reached the thresh hold for Unfair Dismissal and Employment Protection, but you still need to take them on and do your paperwork correctly otherwise you are stepping on egg shells. If they have been there less than three months and thier current contract allows for a probationary period then it is pretty easy but again make sure your paperwork is right.
If they are a poor cleaner, there is a good chance the last company did not do thier part in making sure she is being counseled and disciplinary procedures. This will be your job especially if she has been there more than 2 years (or he). I have found in the past that some companies dont want the hassel but once you start doing the necessary steps, the cleaner either leaves or comes around and starts doing what is expected. Either way you look like the Hero, you get someone to work like they are supposed to or you manage them out, and sometimes when a cleaner is used to a free ride and someone comes in and makes them work, they just leave.
Hope it helps but as I said earlier, seek professional advice on the matter.
Gilbert