There is no supported way to transfer library content synced to an iDevice to another iDevice. You also cannot sync library content from an iDevice to a different iTunes library. Content transfer is from the itunes library on the computer to the iDevice.
The one exception is purchased content (from the iTunes store), which can be transferred to the computer by selecting the "Transfer Purchases" menu option with hte device connected to the computer.
The only resolution I could offer, is that you "Could" (assuming you still have the original computer with the content) transfer the music via homesharing or another method listed in the following KB article:
For transferring music you could use 'PodTransfer App'. I'm the developer of this universal application (iPhone/iPad) that let's you transfer music from your iPod touch or computer directly to another iPhone or iPod.
The app is currently in the Top 25 iPod apps in the photography category.
I hope you find it useful.