I'd generally not bother to recommend a specific printer - nobody can, really, since printers tend to be something you buy and keep for a longish time, so the only thing people can comment on reliably is the one they own and use. (And when they've changed printers, their previous printers are likely to be out of date and no longer on the market).

I'd normally start printer advice with two questions: How much or how frequently do you print? And what sort of things do you print? You've clearly already answered one of those questions ("It's been hammered" and "the rest of the family uses it") - and I'd extrapolate from that, and assume photographic printing is included, possibly in the homework depending on what it is.

The reason I ask these questions is because there are two main types of printers to consider: inkjets and lasers.

Inkjets tend to be cheaper to buy, and the ink cartridges are cheaper, but overall they are more expensive per page. For photographic printing, though - especially with decent quality paper designed for the purpose - lasers can't touch them on quality (though you can still get pretty good results from a laser, which is often more than good enough for the purpose).

Lasers, on the other hand, are more expensive to buy, and toner cartridges are more expensive than ink cartridges - but that toner lasts so much longer, resulting in a much cheaper cost per page.

Also, depending on your requirements, bear in mind the type of paper (etc) you can feed through each - colour lasers tend not to have a straight paper path, so they're more limiting in what you can feed through them without causing problems (most standard label stock is out, as is paper over certain weights, and so on).

So, think about your requirements and consider whether ink or laser is more suitable - if you're printing a lot (as you say) but you don't need photographic quality prints, then you've probably been spending a fortune on ink; that cost could potentially go down. (Unless your kids suddenly start printing more because the "ink" doesn't run out so quickly!)

As to your other requirements:

You use your printer for faxing at the moment; consider if you really need to. How frequently do you fax? Would it be cheaper, perhaps, to use a fax to email service? (I pay £10 plus VAT per year, and then 10p for each page I send - by email, which is faxed to the recipient. Faxes that come to my fax number are free, and are received by me as emails. For the number of faxes I actually send and receive, that's a lot cheaper than wasting money on a line, and extra money on hardware to make use of it. All I need is a scanner - whether it's in the printer, or separate.)

And you mention wireless because everyone uses the printer - but if you are able to put the printer near your router, then you might be able to get away with one that can is networked, but not wireless - i.e. it can be connected to the router with an ethernet cable.

Edit: What I forgot to add was that since you're clearly happy with how the HP Photosmart has performed, and if you're happy with the cost of running it (so the points about lasers being cheaper per page aren't relevant) then HP's latest offering in that range would seem logical!