There seems to be more to this than meets the eye. If true however, the claims about rat infestations and excrement smeared beds warrants a closer look. However, playing devil's advocate:
The list of handouts include: £140 a week child benefits,
Available whether they work or not.
£120 a week disability living allowance,
Available to those who qualify whether they work or not.
£250 a month carers allowance for Peter,
Available to those who qualify regardless.
£527 a week in tax credits,
Available whether they work or not; the name merely changes between Child Tax Credit and Working Tax Credit.
£30 every fortnight for income support.
No arguing that one but it appears aside from their rent (Housing Benefit) the £30 is the only thing they're claiming they wouldn't get if they worked.
If we are to believe the Daily Fail then the father was a member of the Army until he left to care for his disabled wife in 2001. If she is in the 'high rate care' category then with so many children in addition to the burdens of full time care it doesn't seem hard to imagine how things swiftly got out of hand for them. Not to excuse their apparent behaviour, but I can certainly see they might be on a slippery slope.
Whether that excuses having ever more children and allowing things to progress this far is another matter, but I don't think it's as clear cut as the DF makes out (shock, horror!). It sounds to me like they need help, and perhaps a word of advice about contraception, rather than mere condemnation. As I said I'm certainly not outright defending them, but I'd be loathe to make snap judgements or be jumping on the bandwagon without further information we'll likely never have access to.
As Andy said if they are genuinely in need I'd much sooner they had the money they needed than certain others (whether it be hate preaching Mullahs or fat cats scamming the system). JMHO.