Did you download the latest iTunes from here
and install it?
Can you specify what model iPods DO work and what model iPods do NOT work? This document shows all the iPod models
For the 2nd gen shuffle, if you also have a 3rd or 4th gen shuffle, make sure you are using it with the original cable (that has the mini-dock). The USB cable for the 3rd and 4th gen shuffle has a different connector, so it does not work with the 2nd gen shuffle.
If your "older iPod" is a 3rd gen "classic" iPod, it needs a FireWire connection (not USB) to work properly.
As mentioned in previous reply, a 2nd gen iPod shuffle does not work with the USB cable from 4th gen shuffle, because the connector is different. You'll need to get one that is for a 2nd gen shuffle. You can get a less expensive third-party replacement on eBay or Amazon, but you need to make sure it is described as being for the 2nd gen shuffle. This is the official Apple replacement part
The other iPod sounds like a 4th gen "classic" iPod. It should work with a standard USB docking cable. Did it recently work and connect properly with another computer? When you connect it, does it charge? When disconnnected, does it work properly as an iPod (display its screens and play music)?
You can try connecting it to your computer, running iTunes, and (while it is connected) doing a Reset
If that does not work either, you can try putting it into Disk Mode
then connecting it to your computer, to see if that make any difference in it being recognized.
The iPod itself seems to work (as an iPod), and it must charge when connected, or else it wouldn't continue to work.
You said that "all other ipods work," so the new computer's USB port seems to be working fine, and the iTunes installation is working fine (and it's the latest version).
You tried more than one cable, so that docking cable is probably fine (especially if it works with one of the other iPods).
So, I think you have narrowed the cause down to iPod's docking port. My guess is that the "pins" on the port used for data transfer have become faulty. The ones used for charging still work.
Do you still have the old computer, or another computer that runs iTunes? If the iPod also fails to connect and be recognized there (while another iPod using the same connection works), that would be confirmation. If the iPod works on a different computer, then that's obviously not the cause.
One other possibility involves how Windows handles "drive letters." Being mostly a Mac user, I don't fully understand how Windows does this (assigning drive letters), but you can look at this document to see if it is helpful