It could be a bad (new) battery, a bad screen, a bad dock connector, a board issue, or any combination of the previous.
- Bad screen can be ruled out if you can't see the iPod in iTunes or have no image but hear a repetitive "pong" when iDevice is connected to a charger. Let's rule it out for the moment since your device is altogether not detected by iTunes.
- Bad new battery is easy to figure out, since you got so far. Test the voltage usins DC on a multimeter on the two pads of the battery where you soldered it to the board. As a rule of thumb, if you have two pads, measure those. If you have more than 2 pads or contact points, measure on the two furthest. For a Samsung battery it's easy to figure out, the battery is marked with + and - so you know which to measure out of the 3 or 4 contact points. The voltage should be equal to the voltage stamped on the battery AT LEAST or slightly more otherwise it means the battery is flat. For example, a 3.65V on an iPhone battery means the battery is live but flat. 3.85V or more means the iPhone can start from it already.
- Dock connector: if you determine that your battery is OK, try to examine the dock connector under magnification. Are there any broken pins where you plug the cable; is it dirty, corroded? How about where the pins connect to the board? Anything broken? If all seem ok, you can measure the voltages on those pins with the battery connected or not, and with the charging cable plugged in. You must be extremely careful not to short pins (or any 2 components on the board) together with the probes of your multimeter. They should be thin/pointy. Some solder needles to the tips; others file them; others just buy fine probes.
- Next, the board: examine the board for any funky looking component: anything torn from it's pads, damaged, missing? It can be a board issue without being visible pry damage, such as a shorted component preventing the board from starting up, or a dead chip or component preventing the circuit from being complete.
I advise you to get this or a similar device to measure the current draw during charging. It's a great tool for diagnosis of portable devices, cables and chargers. It can tell you how the device is reacting or not to being plugged into a charger, before and after replacing a battery, etc.
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