Hi Martin
If the 'role' still exists when you start the contract then you will need to transfer the person with full continuity of employment - same wages, holiday entitlement etc. It is seen in the eyes of a tribunal that if your new customer had a problem with the cleaner then they should have got the old company to have the person removed under SOSR (Some Other Substantial Reason). This will be open to your client when you take over but i would give her the chance, I find that the sites I have transferred people on, they have been treated badly, wages not always paid on time, never get stock etc. If you go in positive they will either tow the line or not show up on day 1!
Regards
Tim