Ipod touch blink/flash white

May seem like a silly question, but did you changed SSID names when you switched to the new router? If so, make sure you connect both iDevices to the new network's name and reconfigure the Remote app.

Hope this helps!
Thomas

Edit: Are you able to connect to a website from your devices on the new network? And do you still have the old router broadcasting? I ask because maybe your devices are still connected to the old network, but your PC is on the new one. If you haven't already, try going to Settings >> Wi-Fi, and verifying that the network you're connected to is the same that your PC is on. If so, and it's still not working, reset your network settings and see if that helps

It sounds like the home button still works, which is the only button you need to have iTunes recover the iPod for you. Note that recovering the iPod will wipe all of its data completely.

I tried this with an iPhone which I believe was in much worse shape than this and it worked. In my case my phone never got past the Apple logo and would shut down shortly after. Even a hard-drive restoration company wasn't able to access the data, yet the recovery through iTunes did work, so there is hope.

Apple calls this recovery mode.

All you need to do is press and hold the home button on the iPod while connecting it to the PC. iTunes should detect an iPod in recovery mode and allow you to recover the device (again, this wipes out all data completely).