Mini itx build

The route to go will kinda depend on your plans in general. A first gen i5 and a HD5870 are still fairly solid for gaming, but they are both pretty old and starting to become increasingly obsolete if you want to run modern games.

There are ITX boards for socket 1156, but they will be fairly hard to get hold of, lacking in modern features, and depending on your future plans, a huge issue if you want to upgrade. I'd definitely recommend going with a modern motherboard as it does give you a lot more flexibility, the issue is that you are replacing a quality (but ageing) CPU with a fairly budgety one.

Also thanks for your help!

The 16Gb memory should be reusable, depending on it's configuration (2 or 4 sticks), but that shouldn't be an issue.

I'd be a little sceptical about the PSU. These do age, and thermaltake do have some rubbish in their lineup, so they aren't the most trustworthy brand in PSUs. If it's a lower quality, is more than a few years old, or has had a fairly hard life, it might well be something you want to consider replacing.

A HD5870 will fit in both an Elite130 or a Bitfenix Prodigy. In both cases, the card sits against the side of the case, with it's own vented section. It's quite possible you'll get better temps in these cases than you would in an ATX tower.

The Elite130 and the Prodigy don't really compete directly, despite both being ITX. They are very different sizes, with the Prodigy being pretty large (wider than most ATX towers) and offering lots of options for water/air cooling, good airflow and drive bays, while the Elite130 is compact and great value. If you want something easily portable, the Elite 130 is a very clear winner.

If this is a system you plan on spending money on in the future, then it seems to me that setting up a solid foundation by getting a modern motherboard+PSU is a good way to spend your money now. The site you linked doesn't have a fantastic selection but based on the (few) options there, I'd suggest something like
Elite 130 - $70
H87 motherboard - $150
New PSU (on that website it's probably a Corsair CX, as it's cheap, but they aren't terribly good) - $85
Which leaves you with under $100 for a CPU, so you are looking at a Pentium dual core.
It might not seem fantastic (and the part selection doesn't help) but it does mean you have a system which is very upgradable, and you can drop in a modern graphics card and i5 at any point and have a really high end machine.

In theory there should be another option, which is to just make the most of the money you have now, but I'm kinda struggling to figure out what that is. Finding an ITX board for your current CPU is by no means out of the question, but I can't see any for sale, so it's likely to come on eBay or something. That could get the cost down, but it is a bit of a dead end.