As suggested in the Nexus 10 Troubleshooting page, problems with the screen or digitizer can be remedied by replacing the display assembly. Follow this guide to replace the display.

As suggested by the Nexus 10 Troubleshooting page, there are two main ways to solve this problem. Try turning off Location Services in Settings. This is the cause of odd behaviors like these for some users. If this does not work, try updating Android in Settings > Update Software > About tablet > System updates > Check for update. This often solves most software problems with a Nexus.

Most often, problems like these are caused by simple issues. As seen on the Nexus 10 Troubleshooting page, WiFi connection issues are most often caused by simple problems. Make sure that WiFi is on and Airplane Mode is off. Either of these would stop you from connecting to a network.