First, you will of course need to be rooted. I used the method outlined on www.rootkindlefire.com, but others have used Super 1 click, or Zergrush. Whatever floats your particular boat.
Second you'll need the Swype.apk, and a root explorer type app.
Once you've got all that, go ahead and install Swype normally, then navigate to data/app and move the Swype.apk to system/app. Go to a text field and long press to get the input method popup and select Swype, or use Nook Tools app (Swype should start force closing), select kindle keyboard again if you like and reboot, or just reboot (without reselecting Kindle keyboard rebooting takes a while but will happen, just wait). Once you've rebooted, install Swype again. It will tell you it's a system app but your data will be saved blah blah. Once that's done, you should be good to use Swype without issue.
If you want, you can then go ahead and unroot and it should be good. Actually, the only reason you need root is for when you move the apk to system/app. After that, all should work with or without root.
If you get more force closes, just reinstall Swype again. This worked for me on two different kindles so far, so I'm confident, as long as you follow the instructions, it will work for you too.
Sent from my Digital Brick
Edit: fixed Swypeisms