I'm not going to comment on whether you should report your cousin based on how you feel about her, but here's the legislation:
^Circumstances in which a person is to be treated as not liable to make payments in respect of a dwelling
9.?(1) A person who is liable to make payments in respect of a dwelling shall be treated as if he were not so liable where?
(a)the tenancy or other agreement pursuant to which he occupies the dwelling is not on a commercial basis;
(l)in a case to which the preceding sub-paragraphs do not apply, the appropriate authority is satisfied that the liability was created to take advantage of the housing benefit scheme established under Part 7 of the Act.^
As I understand it, the landlady may have written a tenancy agreement stating rent is £700 (for example)per calendar month but isn't actually asking your cousin to pay that amount. The landlady and your cousin are saying that amount of rent is charged in order for your cousin to claim housing benefit which is paid directly to your cousin and there is no suggestion that she pays any money to the landlady in respect of rent.
The local authority would look at whether the landlady would evict your cousin if rent or housing benefit wasn't paid, would want evidence of rent paid to the landlady, eg rent book, receipts, bank statements. Given what you've said and that the landlady is paying school fees, I doubt she'd evict your cousin.
As far as the earnings from nannying and jam selling are concerned, if your cousin has declared the income on her benefit claim, then there's no problem. If she hasn't, then it's fraud.
The OP stated that the cousin "is entitled to the benefit" which confuses me somewhat. Housing benefit is paid to cover actual housing costs, ie; rent. if she's not being charged rent then there is no entitlement. Equally, depending on her income, she may or may not be entitled to housing benefit.
It seems to me from what the OP is saying, that there are two types of possible fraud here. I'm not having a go at anyone because I know the decision to report someone you actually know is a difficult one, but I don't quite understand why some people think benefit fraud is acceptable.