Q: my old computer does not work, how do I get my songs transfered to new computer

If the songs were purchased from iTunes, you can redownload iTunes purchases for free on the new computer using the same Apple ID they were purchased from.

Downloading past purchases from the App Store, iBookstore, and iTunes Store

Make sure to authorize the computer first >  Authorize your computer in iTunes - Apple Support

Otherwise, without a backup to restore from I'm afraid you're out of luck.

See Recover your iTunes library from your iPod or iOS device.

tt2