My 4 year Kindle won't stay connected to wifi.

Make sure that the Airplane mode have not been activated at all . Confirm that your device has the latest software updated or update version has been recently installed . Reset your Kindle and any network devices like modems or routers. Attempt to try to reconnect your Kindle to Wi-Fi again or add a network manually.

Make sure that your kindle Wi-Fi settings is correct. First power reset the Wi-Fi modem/router, then your Kindle

If you want to resolve the issue of wifi in kindle then follow given the link:
How To Resolve The Issue Related To Wi-Fi Connection Through Amazon Kindle...

You can also take kindle help by making a call on kindle customer service number @ +1(844)-305-0086 (Toll-Free)

If book is downloaded on the Kindle Device then do not need you the Wifi. If not THen you need the wifi to connect with the Kindle Server. Kindle Technical Support Registration 877 690 9305

trouble connecting my new Kindle to my wi-fi Mary, Check the attached links,instruction and guides to get kindle connected, Good luck
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Check:
https://www.amazon.co.uk/gp/help/customer/display.html?nodeId=201540590

It is possible that your Kindle Fire's battery has died. If you noticed the battery life diminishing before it wouldn't turn on at all this may be the case. Here is a site that has replacement batteries for the Kindle Fire:
3555A2L Battery for Amazon Kindle Fire D01400

well maybe did you change your wifi password because that happend to me and then i rembered i changed the password for my wifi i hope this helped if not im sorry .

what message do you get when you try to connect to the wi-fi?