A bit late to this thread cos I have been away, but I'd love a good cleaner once a month cos otherwise much of my house gets a lot less. First, it is too big now (DH will not move), second DH is semi-disabled now but was never any good at cleaning (he doesn't see dirt or the point of cleaning and is grossly untidy), and third I appear to have low standards compared to many of you - and I am not saying you are too fussy, just that I am pretty lax.
BUT, in my are there simply are NO cleaners with space available, and few agencies (who all charge the earth) so I have no choice but to struggle myself so once a month would be a great improvement.
I don't have a great issue with paying cash cos it is legal tender and whether or not they declare it for tax purposes is up to them - and I guess that many of them earn less than the tax level anyway. But yes, they ought to pay their self-employed "stamp" to secure a pension, so it would have to be declared. Voluntary contributions cost a lot more.
For me, this young lady would be absolutely fine. Send her to me if you decide to let her go!