My wife's Transformer won't charge either. Tried multiple chargers and nothing. Weird thing is she can plug it into the dock and it charges. Only way she can do it. PC won't recognize it either.
Good chance your port is bad. Try it in the keyboard dock if you can and see if it charges.
Aslo it may be broken sync-charge cable. Try other one. I had similar problem with sync (tablet didn't recognized by PC), that was solved with new cable
The PC recognizes the TF, but not charging...and the dock doesn't charges as well...now I have EOS 4 ROM on the TF and in the status bar it shows, that it's charging but in Menu>Battery it shows that it isn't...and it isn't...it only charging in APX mode, but after a whole night on the charger it is on 22%...WTF? Any helps appriciated...
Another vote on broken cable, I've had thhe same issue, was resolved with a new cable, seems to be something up with them, they seem last about a year and half and break.