Hi Sarah
There is no quick and easy answer to this as I am sure you have established TUPE is not straightforward.
Your sister if terminating the contract will need various pieces of information surrounding the incumbent cleaner's employment T&C in order for an incoming contractor to quote and for a TUPE transfer to take place. This could include requesting to see the cleaner contract of employment. However, the contractor may not agree to share this, but worth asking. It is worth noting though, that a written contract of employment may not negate the TUPE obligations.
Whilst, you can probably terminate the contract with the current contractor (depending on the contract between them and your sister) on the grounds of poor performance. However, as there is no direct legal relationship between your sister and the cleaner, it would follow she cannot terminate the cleaner's contract.
If the transfer did take place and if it was subject to TUPE, which on a 5 hour per week contract is not necessarily a given, there are lots of things you can do to resolve the cleaning quality and if necessary to terminate the cleaners employment, just need to follow the right steps. If you need further advice please email me direct.
Hope this helps
Nick