This is where the SE confusion has really come to a head.
Technically, by any standard, a cleaner working via an agency is employed by that agency. The agency can't just be an introducer and claim an ongoing fee. I know they all do it but maybe this crisis will mean a proper review of all these practices and better governance. Basically, people are being taken advantage of by these firms.
So, if they say they don't employ her (regardless of whether they should) then no, they can't furlough her.
So she should have claimed the SE support. It may not be"much" but it's the same as a lot of people are having to live on! (As a % of income).
If they can't claim the SE support due to not having done tax returns, I'm afraid my response to that is that's their problem, not yours! They made that choice.
What you could have done would be to pay a smaller "retainer", designed to help top up what they will get.
But I assume if they were claiming benefits they weren't entities to while working then they are probably continuing to do so.
Personally, I would not have this person working in my house now and would not use that agency as they clearly know about it. But I am very by the book, I won't pay traders cash to avoid VAT etc, never.
I do know his hard it to get cleaners though! Can you take her on without the agency, cut out their fee, pay her a bit more, tell her she has to declare it (obviously you can't check this unless you actually employ her) - I mean, it's not like the agency is likely to sue you is it!