Try:
- Reset the iOS device. Nothing will be lost
Reset iOS device: Hold down the On/Off button and the Home button at the same time for at
least ten seconds, until the Apple logo appears.
- Reset all settings
Go to Settings > General > Reset and tap Reset All Settings.
All your preferences and settings are reset. Information (such as contacts and calendars) and media (such as songs and videos) aren’t affected.
- Restore from backup. See:
Restore your device from an iCloud or iTunes backup - Apple Support
- Restore to factory settings/new iOS device.
If still such short life, make an appointment at the Genius Bar of an Apple store since it appears you have a hardware problem. Likely a ba/weak battery.
Apple Retail Store - Genius Bar
If only the battery
Apple will exchange your iPod for a refurbished one with a new battery for $79. Otherwise the exchange price is $99 for 16 GB 5G and $149 for the other 5Gs. They do not fix yours.
Service Answer Center - iPod
A third-party place like the following maybe less. Google for more.
iPhone Repair, Service & Parts: iPod Touch, iPad, MacBook Pro Screens
Fix yourself if you are up to it. A replacement battery cost about $15
iPod Touch Repair – iFixit