How should I judge the impact of giving $5k to a family of three kids and two mentally ill parents?

Can other members of the household be questioned about their perception of the recipients' wellbeing before and after?

IMO it'd be ideal to ask them an open-ended "what changed about these people" to avoid priming effects.

However you choose to measure, be sure to check for any negative consequences of the donation. In adults, especially with mental health struggles, you may note resentment about how the aid was delivered or disappointment that it wasn't more.

It would be interesting to A/B test the impact of spending a lump sum like this, versus splitting the same sum over several weeks, months, quarters, or years. Part of experiencing a sense of security is knowing there'll be more where it came from -- so for instance it could be the case that giving a kid the interest from $1k as a monthly allowance might have greater positive overall impact than just giving them $1k worth of items all at once.

Do you have a good idea of what the situation looks like currently, what the case worker will buy, and what the situation looks like after it's been bought?

If you can extrapolate how the situation would look if you didn't make the intervention, and you know what the intervention did, you could try to compute the QALY difference between those two estimates to get an impression of its effectiveness. Remember to beware of regression to the mean, though.

Without this sort of detailed knowledge, I don't think it's possible to estimate the effect.

It's going to be tricky. You may already be too close to the situation to judge impartially, and a case study is going to be difficult to use as evidence against population-level surveys of well-being, especially for your implied time horizon. You could attempt to benchmark against previous work, e.g. see what the literature has to say about the effects of poverty on diet, educational attainment, etc. in first-world cities, but your one new data point still won't generalize and it wouldn't be doing the heavy lifting in your argument for localism at that point.

With a goal to "alleviate immediate and preventable suffering," QALY seems to be a pretty terrible metric. You need to measure immediacy, preventability, and suffering, or at least the suffering due to just the immediate and preventable causes. I would suggest suffering needs construct definition before you consider an operationalization. 

It would be smart to measure pre- and post-intervention. The good news is that if suffering is a subjective psychological state, you could do post-only and measure perceived change. If you're worried about self-report, you could do observer-report (case worker), but since they'll be doling out the funds that could be biased as well (and presumably they'll be basing these decisions in part on self-reports of what is a "fire").

The type of counterfactual analysis tailcalled suggests is likely what the family or case worker would be mentally approximating when responding to how big of a difference paying off the utility bill or a pizza night made to their suffering. Plus, n=1, so a qualitative assessment may be all you can really do - if they rate something a 7/7 on reducing their suffering and everything else came in at 1-2, sure, that hits you between the eyes, though you'd probably get that anyway from them saying "the most important thing, by a mile, was X, and this is why..."