Your old machine is compatible with wi-fi, but to use that you would need a dongle - I've used one in the past on a desktop older than yours.
The first thing to do is to have a look at your router ans see if there are any stickers on it. What you are looking for is one which mentions SSID and Passphrase (or similar), and make a note of both of them.
If no stickers are on the router, then follow Methyl's advice and let us know the make and model of the router (they will be on it somewhere), then someone can help you set up security on the router and tell you how to connect the Kindle to it.
The next part is a bit general. I don't have a Kindle, though my wife does have a Fire HD but, as I am very attached to various bits of my anatomy, I make sure I don't touch it.
If you do find a sticker with the SSID and Passphrase on your router, start your Kindle and at the top of the screen you should see a number of icons. One is a cogwheel (or it might look like a circle with a hole in it if your eyesight is as good as mine). Tap on that to get into the quick settings menu.
One of the symbols on that will be labelled Wi-Fi - tap on that. It should bring up a screen which shows wi-fi connectivity and any available wi-fi connections.
If yours shows that wi-fi is off, touch the On part of the screen. Then look down the list of available networks. Find the one which matches the SSID on your router and touch that. if it comes up with a question asking if you want to connect to it, the answer is yes. If nothing happens, touch the one for your router and leave your finger there for a couple of seconds.
When it asks for the passphrase, enter the one you noted on your router, and that should then allow you to use your wi-fi connection.