Game Center Not connecting

Some things you can try. Do you have access to another WiFi network where you can try? That has worked for several people.

Try a restart.

Restart your iPhone   *

Restart your iPad.   *

Try a forced restart. See the section towards the end of the article.

Reset Device. -

If necessary

Hard reset.     -

Do a backup and try a factory reset. Then restore from the backup.

Restore to Factory setting.      *

Restore - If you see an error when you update or restore your iPhone, iPad, or iPod.   *

Backup - Restore your device from an iCloud or iTunes backup.     *