Watchmaker's post is a good place to start.
Some unorganized thoughts:
-It's a neat idea but curiously (and unsurprisingly), there's no discussion on scalability. This is presumably a fragile matrix and I would imagine that this polymer is going to tear or disintegrate under its own weight after your first water cycle if you were to handle it. If this were to be developed for use in real-world conditions, I'd bet that it would be affixed to a platform that would not inhibit too much water uptake but would also allow for it to be transported easily.
-Scaling also brings the issue of retrieving the water. It's not like you can just whip this out and have a cup of water in an hour. You'll need a means of warming the gel to remove the water, and then a way to collect that water.
-You might run into trouble purchasing these materials. Sigma's not always too keen on sending laboratory grade chemicals to residential addresses, YMMV.
-From a practical perspective, you probably don't need a vortex mixer if you're only making a small amount at home - you could get away with just vigorous shaking in a sealed container.
-However, you will need a scale that's more accurate than your run of the mill kitchen scale. A really nice postal scale might suffice, but you'd be better served by a really cheap lab grade model.
-I don't know off the top of my head what the pH of 'Solution A' should be, but its pH is going to be key to creating a functional gel. pH strips would probably be okay-ish enough to give you a close idea of where your pH is without having to buy a meter.
-Without practice, there's a really good chance that a DIY version of this would yield less than half of the lab-scale version.
-Don't forget that you need to further purify the water.
This is honestly your best shot for trying - any lab at any level will have a balance, vortex mixer, access to acids and bases, and most importantly - someone on hand who can help make sure you're doing what you're supposed to be doing:
The equipment is slightly specialized, but any university chemistry lab should be able to do this with ease, maybe make a friend at your local school if you want to actually synthesis some?