there are no phone jacks in the house, just RJ45 ports
I know you said POTS is unneeded, but you can plug RJ11/RJ12 plugs into RJ45 jacks. You could also rewire a cable with an RJ45 plug on one end if you were concerned with bending the jack's outer pins.
I can't tell where the individual wires from the various jacks come out, but it doesn't appear like they are connected - does that make sense?
No, I don't follow.
there's a block on the wall where the used Cat5 connections come in, and it is mostly unused
A punchdown panel, more-or-less like
this? (There are many different sorts.)
Maybe a picture would help?
Generally speaking, in a setup like this, you'll have all of your jacks throughout the house terminated in the closet. It sounds like maybe the cables are run to the closet, but they're not actually terminated anywhere?
If that's the case, and you do have a punchdown block like I linked to above, you'll need to get a punchdown tool to terminate them. Usually, you terminate every jack, but you might not have every jack patched into a switch (or other communications device).
Then you generally have your switches and whatnot mounted along side that patch panel, and run short cables between the jack terminations and the device you want to plug into.