While following the instructions at https://www.amazon.co.uk/gp/help/customer/display.html?nodeId=201730640 I still could not connect a family Kindle to the Internet.
What was happening was that the Kindle HD found the correct network to connect to, the correct password was being entered into the device from the rear of the Plusnet Hub, the Wifi on the Kindle settings was showing 'Connected' then briefly 'No Internet'. I suspect an app may have got its knickers in a twist and been closing the connection as soon as it was made - I'm really not sure. So to get around the unknown cause ....
... it was RESOLVED.
This is a last resort, as you lose all settings, and downloaded apps.
Before starting, you will need the registered Amazon email address you use for the Amazon account, and the associated password.
Press the power button and the volume up switch together for up to 40 seconds.
The device will start to reboot.
The device stops with some text on the screen.
Use the sound up/down buttons to navigate to the required option (in this case 'reset device'), and the power on/off switch to launch that option.
An orange screen comes up and you then have to select country and language options, followed by Amazon email address and password in order to register the device.
You will need those to next go to the Amazon store to download the apps you want on the device.
If you are unsure what apps you had on the device, Amazon will have kept a record of the most recent apps downloaded in the last 6 months.